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

Percentage Accurate: 85.6% → 99.2%

Time: 8.6s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. associate-*r* N/A

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

   9. mul-1-neg N/A

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

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

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

   11. associate--l- N/A

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

   12. lower--.f64 N/A

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

5. Applied rewrites 99.0%

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

Alternative 2: 90.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-99} \lor \neg \left(t \leq 3.8 \cdot 10^{-115}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.25e-99) (not (<= t 3.8e-115)))
   (fma (log y) x (- t))
   (- (* (log y) x) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.25e-99) || !(t <= 3.8e-115)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = (log(y) * x) - (z * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.25e-99) || !(t <= 3.8e-115))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = Float64(Float64(log(y) * x) - Float64(z * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.25e-99], N[Not[LessEqual[t, 3.8e-115]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.24999999999999992e-99 or 3.79999999999999992e-115 < t

   1. Initial program 91.7%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 90.6%

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

   if -1.24999999999999992e-99 < t < 3.79999999999999992e-115

   1. Initial program 73.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate--l- N/A

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

      12. lower--.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 95.5%

      \[\leadsto \log y \cdot x - z \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-99} \lor \neg \left(t \leq 3.8 \cdot 10^{-115}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-83} \lor \neg \left(x \leq 1.7 \cdot 10^{-87}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y, z, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7.6e-83) (not (<= x 1.7e-87)))
   (fma (log y) x (- t))
   (fma (* (fma -0.5 y -1.0) y) z (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7.6e-83) || !(x <= 1.7e-87)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = fma((fma(-0.5, y, -1.0) * y), z, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7.6e-83) || !(x <= 1.7e-87))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = fma(Float64(fma(-0.5, y, -1.0) * y), z, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7.6e-83], N[Not[LessEqual[x, 1.7e-87]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.59999999999999953e-83 or 1.6999999999999999e-87 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 91.9%

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

   if -7.59999999999999953e-83 < x < 1.6999999999999999e-87

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{neg}\left(t\right)}\right) \]

      13. lower-neg.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 92.4%

      \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot y - 1\right) \cdot y, z, -t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 92.4%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 92.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-83} \lor \neg \left(x \leq 1.7 \cdot 10^{-87}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y, z, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.62 \cdot 10^{+66} \lor \neg \left(x \leq 0.003\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.62e+66) (not (<= x 0.003)))
   (* (log y) x)
   (- (* (fma (* z (fma -0.3333333333333333 y -0.5)) y (- z)) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.62e+66) || !(x <= 0.003)) {
		tmp = log(y) * x;
	} else {
		tmp = (fma((z * fma(-0.3333333333333333, y, -0.5)), y, -z) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.62e+66) || !(x <= 0.003))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(fma(Float64(z * fma(-0.3333333333333333, y, -0.5)), y, Float64(-z)) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.62e+66], N[Not[LessEqual[x, 0.003]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(z * N[(-0.3333333333333333 * y + -0.5), $MachinePrecision]), $MachinePrecision] * y + (-z)), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.62e66 or 0.0030000000000000001 < x

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. remove-double-neg N/A

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

       3. mul-1-neg N/A

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

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

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

       5. log-rec N/A

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

       6. lower-*.f64 N/A

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

       7. log-rec N/A

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

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

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

       9. mul-1-neg N/A

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

       10. remove-double-neg N/A

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

       11. lower-log.f64 79.0

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

   11. Applied rewrites 79.0%

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

   if -1.62e66 < x < 0.0030000000000000001

   1. Initial program 76.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. cancel-sign-sub N/A

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

      9. mul-1-neg N/A

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

      10. lower-log1p.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{neg}\left(t\right)}\right) \]

      13. lower-neg.f64 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 83.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.62 \cdot 10^{+66} \lor \neg \left(x \leq 0.003\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.4% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. mul-1-neg N/A

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

   10. lower-log1p.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{neg}\left(t\right)}\right) \]

   13. lower-neg.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right), z, -t\right) \]
7. Step-by-step derivation

8. Applied rewrites 55.2%

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

Alternative 6: 57.4% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. mul-1-neg N/A

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

   10. lower-log1p.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{neg}\left(t\right)}\right) \]

   13. lower-neg.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 55.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 55.2%

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

Alternative 7: 57.4% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. mul-1-neg N/A

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

   10. lower-log1p.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{neg}\left(t\right)}\right) \]

   13. lower-neg.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 55.2%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot y - 1\right) \cdot y, z, -t\right) \]
9. Step-by-step derivation

10. Applied rewrites 55.2%

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

Alternative 8: 48.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-99} \lor \neg \left(t \leq 1.46 \cdot 10^{-111}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.25e-99) (not (<= t 1.46e-111))) (- t) (* (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.25e-99) || !(t <= 1.46e-111)) {
		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 <= (-1.25d-99)) .or. (.not. (t <= 1.46d-111))) 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 <= -1.25e-99) || !(t <= 1.46e-111)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.25e-99) or not (t <= 1.46e-111):
		tmp = -t
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.25e-99) || !(t <= 1.46e-111))
		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 <= -1.25e-99) || ~((t <= 1.46e-111)))
		tmp = -t;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.25e-99], N[Not[LessEqual[t, 1.46e-111]], $MachinePrecision]], (-t), N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.24999999999999992e-99 or 1.46e-111 < t

   1. Initial program 91.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.7

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

   5. Applied rewrites 59.7%

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

   if -1.24999999999999992e-99 < t < 1.46e-111

   1. Initial program 74.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

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

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

      11. associate--l- N/A

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

      12. lower--.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 28.8%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{-99} \lor \neg \left(t \leq 1.46 \cdot 10^{-111}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. mul-1-neg N/A

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

   10. lower-log1p.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{neg}\left(t\right)}\right) \]

   13. lower-neg.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 55.2%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot y - 1\right) \cdot y, z, -t\right) \]
9. Step-by-step derivation

10. Applied rewrites 55.2%

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

11. Taylor expanded in y around 0

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

13. Applied rewrites 55.0%

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

Alternative 10: 57.3% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. mul-1-neg N/A

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

   10. lower-log1p.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{neg}\left(t\right)}\right) \]

   13. lower-neg.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 55.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 55.0%

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

Alternative 11: 57.0% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. associate-*r* N/A

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

   9. mul-1-neg N/A

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

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

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

   11. associate--l- N/A

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

   12. lower--.f64 N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 54.8%

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

Alternative 12: 42.8% accurate, 73.3× 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 85.3%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 40.0

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

5. Applied rewrites 40.0%

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

Developer Target 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * Math.log(y)));
}

Python

def code(x, y, z, t):
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * math.log(y)))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
end

Wolfram

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

Reproduce

herbie shell --seed 2025017 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))