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

Percentage Accurate: 89.4% → 99.4%

Time: 13.0s

Alternatives: 17

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(-y\right)\\ t_2 := \mathsf{fma}\left(\log y, x - 1, \left(1 - z\right) \cdot t\_1\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(t\_1, z - 1, \log y \cdot \left(x - 1\right)\right), t\_2 \cdot {t\_2}^{-1}, -t\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (log1p (- y))) (t_2 (fma (log y) (- x 1.0) (* (- 1.0 z) t_1))))
   (fma
    (fma t_1 (- z 1.0) (* (log y) (- x 1.0)))
    (* t_2 (pow t_2 -1.0))
    (- t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log1p(-y);
	double t_2 = fma(log(y), (x - 1.0), ((1.0 - z) * t_1));
	return fma(fma(t_1, (z - 1.0), (log(y) * (x - 1.0))), (t_2 * pow(t_2, -1.0)), -t);
}

Julia

function code(x, y, z, t)
	t_1 = log1p(Float64(-y))
	t_2 = fma(log(y), Float64(x - 1.0), Float64(Float64(1.0 - z) * t_1))
	return fma(fma(t_1, Float64(z - 1.0), Float64(log(y) * Float64(x - 1.0))), Float64(t_2 * (t_2 ^ -1.0)), Float64(-t))
end

Wolfram

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

Derivation

1. Initial program 90.7%

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

4. Applied rewrites 99.7%

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

Alternative 2: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq -5000 \lor \neg \left(t\_1 \leq 660\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (or (<= t_1 -5000.0) (not (<= t_1 660.0)))
     (- (* (log y) x) t)
     (- (- y (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if ((t_1 <= -5000.0) || !(t_1 <= 660.0)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = (y - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))
    if ((t_1 <= (-5000.0d0)) .or. (.not. (t_1 <= 660.0d0))) then
        tmp = (log(y) * x) - t
    else
        tmp = (y - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)));
	double tmp;
	if ((t_1 <= -5000.0) || !(t_1 <= 660.0)) {
		tmp = (Math.log(y) * x) - t;
	} else {
		tmp = (y - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))
	tmp = 0
	if (t_1 <= -5000.0) or not (t_1 <= 660.0):
		tmp = (math.log(y) * x) - t
	else:
		tmp = (y - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if ((t_1 <= -5000.0) || !(t_1 <= 660.0))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(Float64(y - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	tmp = 0.0;
	if ((t_1 <= -5000.0) || ~((t_1 <= 660.0)))
		tmp = (log(y) * x) - t;
	else
		tmp = (y - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5000.0], N[Not[LessEqual[t$95$1, 660.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -5e3 or 660 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 93.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -5e3 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 660

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 86.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 84.9%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq -5000 \lor \neg \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 660\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 4: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 5: 76.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -1.00000002:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;x - 1 \leq 10^{+21}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= (- x 1.0) -1e+94)
     t_1
     (if (<= (- x 1.0) -1.00000002)
       (- (* (- 1.0 z) y) t)
       (if (<= (- x 1.0) 1e+21) (- (- y (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((x - 1.0) <= -1e+94) {
		tmp = t_1;
	} else if ((x - 1.0) <= -1.00000002) {
		tmp = ((1.0 - z) * y) - t;
	} else if ((x - 1.0) <= 1e+21) {
		tmp = (y - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if ((x - 1.0d0) <= (-1d+94)) then
        tmp = t_1
    else if ((x - 1.0d0) <= (-1.00000002d0)) then
        tmp = ((1.0d0 - z) * y) - t
    else if ((x - 1.0d0) <= 1d+21) then
        tmp = (y - log(y)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if ((x - 1.0) <= -1e+94) {
		tmp = t_1;
	} else if ((x - 1.0) <= -1.00000002) {
		tmp = ((1.0 - z) * y) - t;
	} else if ((x - 1.0) <= 1e+21) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if (x - 1.0) <= -1e+94:
		tmp = t_1
	elif (x - 1.0) <= -1.00000002:
		tmp = ((1.0 - z) * y) - t
	elif (x - 1.0) <= 1e+21:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1e+94)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -1.00000002)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	elseif (Float64(x - 1.0) <= 1e+21)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if ((x - 1.0) <= -1e+94)
		tmp = t_1;
	elseif ((x - 1.0) <= -1.00000002)
		tmp = ((1.0 - z) * y) - t;
	elseif ((x - 1.0) <= 1e+21)
		tmp = (y - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1e+94], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.00000002], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], 1e+21], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e94 or 1e21 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.0%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if -1e94 < (-.f64 x #s(literal 1 binary64)) < -1.0000000200000001

   1. Initial program 74.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.8%

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

   if -1.0000000200000001 < (-.f64 x #s(literal 1 binary64)) < 1e21

   1. Initial program 89.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+94}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x - 1 \leq -1.00000002:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;x - 1 \leq 10^{+21}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -1.00000002:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;x - 1 \leq 10^{+21}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= (- x 1.0) -1e+94)
     t_1
     (if (<= (- x 1.0) -1.00000002)
       (- (* (- 1.0 z) y) t)
       (if (<= (- x 1.0) 1e+21) (- (+ (log y) t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((x - 1.0) <= -1e+94) {
		tmp = t_1;
	} else if ((x - 1.0) <= -1.00000002) {
		tmp = ((1.0 - z) * y) - t;
	} else if ((x - 1.0) <= 1e+21) {
		tmp = -(log(y) + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if ((x - 1.0d0) <= (-1d+94)) then
        tmp = t_1
    else if ((x - 1.0d0) <= (-1.00000002d0)) then
        tmp = ((1.0d0 - z) * y) - t
    else if ((x - 1.0d0) <= 1d+21) then
        tmp = -(log(y) + t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if ((x - 1.0) <= -1e+94) {
		tmp = t_1;
	} else if ((x - 1.0) <= -1.00000002) {
		tmp = ((1.0 - z) * y) - t;
	} else if ((x - 1.0) <= 1e+21) {
		tmp = -(Math.log(y) + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if (x - 1.0) <= -1e+94:
		tmp = t_1
	elif (x - 1.0) <= -1.00000002:
		tmp = ((1.0 - z) * y) - t
	elif (x - 1.0) <= 1e+21:
		tmp = -(math.log(y) + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1e+94)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -1.00000002)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	elseif (Float64(x - 1.0) <= 1e+21)
		tmp = Float64(-Float64(log(y) + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if ((x - 1.0) <= -1e+94)
		tmp = t_1;
	elseif ((x - 1.0) <= -1.00000002)
		tmp = ((1.0 - z) * y) - t;
	elseif ((x - 1.0) <= 1e+21)
		tmp = -(log(y) + t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1e+94], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.00000002], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], 1e+21], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e94 or 1e21 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.0%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if -1e94 < (-.f64 x #s(literal 1 binary64)) < -1.0000000200000001

   1. Initial program 74.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.8%

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

   if -1.0000000200000001 < (-.f64 x #s(literal 1 binary64)) < 1e21

   1. Initial program 89.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+94}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x - 1 \leq -1.00000002:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;x - 1 \leq 10^{+21}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -4 \cdot 10^{+19}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x - 1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(-z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x 1.0) -4e+19)
   (- (* (log y) x) t)
   (if (<= (- x 1.0) -1.0)
     (- (fma (- z) y (- (log y))) t)
     (fma (- x 1.0) (log y) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - 1.0) <= -4e+19) {
		tmp = (log(y) * x) - t;
	} else if ((x - 1.0) <= -1.0) {
		tmp = fma(-z, y, -log(y)) - t;
	} else {
		tmp = fma((x - 1.0), log(y), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -4e+19)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (Float64(x - 1.0) <= -1.0)
		tmp = Float64(fma(Float64(-z), y, Float64(-log(y))) - t);
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - 1.0), $MachinePrecision], -4e+19], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.0], N[(N[((-z) * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4e19

   1. Initial program 94.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -4e19 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 86.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.8%

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

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

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 95.5

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

   5. Applied rewrites 95.5%

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

Alternative 8: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 9: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+19}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.2e+19)
   (- (* (log y) x) t)
   (if (<= x 5e-27)
     (- (- (fma (- z 1.0) y (log y))) t)
     (fma (- x 1.0) (log y) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.2e+19) {
		tmp = (log(y) * x) - t;
	} else if (x <= 5e-27) {
		tmp = -fma((z - 1.0), y, log(y)) - t;
	} else {
		tmp = fma((x - 1.0), log(y), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.2e+19)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (x <= 5e-27)
		tmp = Float64(Float64(-fma(Float64(z - 1.0), y, log(y))) - t);
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.2e+19], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 5e-27], N[((-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision]) - t), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e19

   1. Initial program 94.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 94.1

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

   5. Applied rewrites 94.1%

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

   if -2.2e19 < x < 5.0000000000000002e-27

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 96.9%

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

   if 5.0000000000000002e-27 < x

   1. Initial program 95.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 95.8

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

   5. Applied rewrites 95.8%

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

Alternative 10: 67.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+94} \lor \neg \left(x - 1 \leq 5 \cdot 10^{+38}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -1e+94) (not (<= (- x 1.0) 5e+38)))
   (* (log y) x)
   (- (* (- 1.0 z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -1e+94) || !((x - 1.0) <= 5e+38)) {
		tmp = log(y) * x;
	} else {
		tmp = ((1.0 - z) * y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    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 - 1.0d0) <= (-1d+94)) .or. (.not. ((x - 1.0d0) <= 5d+38))) then
        tmp = log(y) * x
    else
        tmp = ((1.0d0 - 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 - 1.0) <= -1e+94) || !((x - 1.0) <= 5e+38)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = ((1.0 - z) * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - 1.0) <= -1e+94) or not ((x - 1.0) <= 5e+38):
		tmp = math.log(y) * x
	else:
		tmp = ((1.0 - z) * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -1e+94) || !(Float64(x - 1.0) <= 5e+38))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x - 1.0) <= -1e+94) || ~(((x - 1.0) <= 5e+38)))
		tmp = log(y) * x;
	else
		tmp = ((1.0 - z) * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -1e+94], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], 5e+38]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e94 or 4.9999999999999997e38 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.9%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 87.4

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

   7. Applied rewrites 87.4%

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

   if -1e94 < (-.f64 x #s(literal 1 binary64)) < 4.9999999999999997e38

   1. Initial program 86.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 55.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+94} \lor \neg \left(x - 1 \leq 5 \cdot 10^{+38}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 1e+207)
   (- (fma (log y) (- x 1.0) y) t)
   (- (* (log1p (- y)) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 1e+207) {
		tmp = fma(log(y), (x - 1.0), y) - t;
	} else {
		tmp = (log1p(-y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 1e+207)
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	else
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+207], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 1e207

   1. Initial program 94.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.5%

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

   if 1e207 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 35.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 79.1

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

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 1e+207)
   (fma (- x 1.0) (log y) (- t))
   (- (* (log1p (- y)) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 1e+207) {
		tmp = fma((x - 1.0), log(y), -t);
	} else {
		tmp = (log1p(-y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 1e+207)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	else
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+207], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 1e207

   1. Initial program 94.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if 1e207 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 35.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 79.1

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

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   9. associate--r+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 14: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   9. associate--r+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 99.3%

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

Alternative 15: 46.2% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   9. associate--r+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 39.3%

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

Alternative 16: 46.0% accurate, 20.5× 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

real(8) function code(x, y, z, t)
    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 90.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(z - 1\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(0 - \left(z + \color{blue}{-1}\right), y, \log y \cdot \left(x - 1\right)\right) - t \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(-1 + z\right)}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   9. associate--r+ N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - -1\right) - z}, y, \log y \cdot \left(x - 1\right)\right) - t \]

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 39.1%

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

Alternative 17: 35.8% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.7%

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

3. Taylor expanded in 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 30.4

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

5. Applied rewrites 30.4%

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

Reproduce

herbie shell --seed 2024325 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))