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

Percentage Accurate: 85.7% → 99.6%

Time: 6.5s

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: 85.7% 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: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.8

       \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

5. Applied rewrites 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lower--.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 2: 76.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y + z \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+123} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+46}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* x (log y)) (* z (log (- 1.0 y))))))
   (if (or (<= t_1 -2e+123) (not (<= t_1 2e+46)))
     (* (log y) x)
     (- (* (- y) (* (fma (fma 0.3333333333333333 y 0.5) y 1.0) z)) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) + (z * log((1.0 - y)));
	double tmp;
	if ((t_1 <= -2e+123) || !(t_1 <= 2e+46)) {
		tmp = log(y) * x;
	} else {
		tmp = (-y * (fma(fma(0.3333333333333333, y, 0.5), y, 1.0) * z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) + Float64(z * log(Float64(1.0 - y))))
	tmp = 0.0
	if ((t_1 <= -2e+123) || !(t_1 <= 2e+46))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(-y) * Float64(fma(fma(0.3333333333333333, y, 0.5), y, 1.0) * z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+123], N[Not[LessEqual[t$95$1, 2e+46]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[((-y) * N[(N[(N[(0.3333333333333333 * y + 0.5), $MachinePrecision] * y + 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -1.99999999999999996e123 or 2e46 < (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 97.7%

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

   3. Taylor expanded in x around inf

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

      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 85.1

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

   5. Applied rewrites 85.1%

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

   if -1.99999999999999996e123 < (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 2e46

   1. Initial program 74.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 80.0

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

   8. Applied rewrites 80.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y + z \cdot \log \left(1 - y\right) \leq -2 \cdot 10^{+123} \lor \neg \left(x \cdot \log y + z \cdot \log \left(1 - y\right) \leq 2 \cdot 10^{+46}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, y, 0.5\right), y, 1\right) \cdot z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-log.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 4: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 99.8

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

5. Applied rewrites 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. associate--l+ N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lower--.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 5: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 6: 90.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-98} \lor \neg \left(x \leq 4.1 \cdot 10^{-128}\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot -0.5 - z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.75e-98) (not (<= x 4.1e-128)))
   (- (* (log y) x) t)
   (- (* (- (* (* z y) -0.5) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.75e-98) || !(x <= 4.1e-128)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = ((((z * y) * -0.5) - z) * y) - t;
	}
	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 ((x <= (-1.75d-98)) .or. (.not. (x <= 4.1d-128))) then
        tmp = (log(y) * x) - t
    else
        tmp = ((((z * y) * (-0.5d0)) - z) * y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.75e-98) || !(x <= 4.1e-128)) {
		tmp = (Math.log(y) * x) - t;
	} else {
		tmp = ((((z * y) * -0.5) - z) * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.75e-98) or not (x <= 4.1e-128):
		tmp = (math.log(y) * x) - t
	else:
		tmp = ((((z * y) * -0.5) - z) * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.75e-98) || !(x <= 4.1e-128))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(z * y) * -0.5) - z) * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.75e-98) || ~((x <= 4.1e-128)))
		tmp = (log(y) * x) - t;
	else
		tmp = ((((z * y) * -0.5) - z) * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.75e-98], N[Not[LessEqual[x, 4.1e-128]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(z * y), $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7500000000000001e-98 or 4.1e-128 < x

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if -1.7500000000000001e-98 < x < 4.1e-128

   1. Initial program 63.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 95.0

         \[\leadsto \left(\left(z \cdot y\right) \cdot -0.5 - z\right) \cdot y - t \]

   8. Applied rewrites 95.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-98} \lor \neg \left(x \leq 4.1 \cdot 10^{-128}\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot -0.5 - z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-128}:\\ \;\;\;\;\left(\left(z \cdot y\right) \cdot -0.5 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.75e-98)
   (fma (log y) x (- t))
   (if (<= x 4.1e-128)
     (- (* (- (* (* z y) -0.5) z) y) t)
     (- (* (log y) x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.75e-98) {
		tmp = fma(log(y), x, -t);
	} else if (x <= 4.1e-128) {
		tmp = ((((z * y) * -0.5) - z) * y) - t;
	} else {
		tmp = (log(y) * x) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.75e-98)
		tmp = fma(log(y), x, Float64(-t));
	elseif (x <= 4.1e-128)
		tmp = Float64(Float64(Float64(Float64(Float64(z * y) * -0.5) - z) * y) - t);
	else
		tmp = Float64(Float64(log(y) * x) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.75e-98], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], If[LessEqual[x, 4.1e-128], N[(N[(N[(N[(N[(z * y), $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7500000000000001e-98

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. associate--l+ N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lower--.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 93.5

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

   10. Applied rewrites 93.5%

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

   if -1.7500000000000001e-98 < x < 4.1e-128

   1. Initial program 63.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 95.0

         \[\leadsto \left(\left(z \cdot y\right) \cdot -0.5 - z\right) \cdot y - t \]

   8. Applied rewrites 95.0%

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

   if 4.1e-128 < x

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 8: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 9: 57.6% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-log.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in z around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 56.3

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

8. Applied rewrites 56.3%

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

Alternative 10: 57.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 56.2

      \[\leadsto \left(\left(z \cdot y\right) \cdot -0.5 - z\right) \cdot y - t \]

8. Applied rewrites 56.2%

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

Alternative 11: 57.1% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-log.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in z around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 56.3

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

8. Applied rewrites 56.3%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 56.1%

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

Alternative 12: 42.9% 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 82.9%

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

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

   2. lower-neg.f64 39.8

      \[\leadsto -t \]

5. Applied rewrites 39.8%

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

Developer Target 1: 99.5% 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 2025064 
(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))