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

Percentage Accurate: 89.6% → 99.8%

Time: 12.2s

Alternatives: 15

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: 89.6% 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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (log y) x (fma (log y) -1.0 (fma (log1p (- y)) (- z 1.0) (- t)))))

C

double code(double x, double y, double z, double t) {
	return fma(log(y), x, fma(log(y), -1.0, fma(log1p(-y), (z - 1.0), -t)));
}

Julia

function code(x, y, z, t)
	return fma(log(y), x, fma(log(y), -1.0, fma(log1p(Float64(-y)), Float64(z - 1.0), Float64(-t))))
end

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. distribute-lft-in N/A

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

   9. associate-+l+ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. metadata-eval N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

Alternative 2: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 76.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x - 1 \leq -4 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot y - t\\ \mathbf{elif}\;x - 1 \leq 10^{+88}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (- x 1.0) -4e+91)
     t_1
     (if (<= (- x 1.0) -5e+16)
       (- (* (* (fma -0.5 y -1.0) z) y) t)
       (if (<= (- x 1.0) 1e+88) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((x - 1.0) <= -4e+91) {
		tmp = t_1;
	} else if ((x - 1.0) <= -5e+16) {
		tmp = ((fma(-0.5, y, -1.0) * z) * y) - t;
	} else if ((x - 1.0) <= 1e+88) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(x - 1.0) <= -4e+91)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -5e+16)
		tmp = Float64(Float64(Float64(fma(-0.5, y, -1.0) * z) * y) - t);
	elseif (Float64(x - 1.0) <= 1e+88)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -4e+91], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+16], N[(N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], 1e+88], 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)) < -4.00000000000000032e91 or 9.99999999999999959e87 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 98.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   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 81.0

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

   7. Applied rewrites 81.0%

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

   if -4.00000000000000032e91 < (-.f64 x #s(literal 1 binary64)) < -5e16

   1. Initial program 72.6%

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

   3. Taylor expanded in y around 0

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

      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 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. metadata-eval N/A

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

      11. sub-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 72.9%

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

   if -5e16 < (-.f64 x #s(literal 1 binary64)) < 9.99999999999999959e87

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 78.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -4 \cdot 10^{+91}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x - 1 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot y - t\\ \mathbf{elif}\;x - 1 \leq 10^{+88}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y - t\\ \mathbf{if}\;x - 1 \leq -1.002:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.999995:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (- x 1.0) (log y)) t)))
   (if (<= (- x 1.0) -1.002)
     t_1
     (if (<= (- x 1.0) -0.999995) (- (- (fma (- z 1.0) y (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * log(y)) - t;
	double tmp;
	if ((x - 1.0) <= -1.002) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.999995) {
		tmp = -fma((z - 1.0), y, log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - 1.0) * log(y)) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1.002)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.999995)
		tmp = Float64(Float64(-fma(Float64(z - 1.0), y, log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.002], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.999995], N[((-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.002 or -0.99999499999999997 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 93.1

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

   5. Applied rewrites 93.1%

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

   if -1.002 < (-.f64 x #s(literal 1 binary64)) < -0.99999499999999997

   1. Initial program 85.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. 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 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.7%

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

Alternative 5: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 99.4

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

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 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-*.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 6: 99.4% 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 90.4%

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

3. Taylor expanded in y around 0

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

   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 99.7

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

5. Applied rewrites 99.7%

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

Alternative 7: 89.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-z\right) \cdot y - t\\ \mathbf{if}\;z - 1 \leq -1 \cdot 10^{+248}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z - 1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (- z) y) t)))
   (if (<= (- z 1.0) -1e+248)
     t_1
     (if (<= (- z 1.0) 5e+280) (- (* (- x 1.0) (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-z * y) - t;
	double tmp;
	if ((z - 1.0) <= -1e+248) {
		tmp = t_1;
	} else if ((z - 1.0) <= 5e+280) {
		tmp = ((x - 1.0) * 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 = (-z * y) - t
    if ((z - 1.0d0) <= (-1d+248)) then
        tmp = t_1
    else if ((z - 1.0d0) <= 5d+280) then
        tmp = ((x - 1.0d0) * log(y)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-z * y) - t;
	double tmp;
	if ((z - 1.0) <= -1e+248) {
		tmp = t_1;
	} else if ((z - 1.0) <= 5e+280) {
		tmp = ((x - 1.0) * Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-z * y) - t
	tmp = 0
	if (z - 1.0) <= -1e+248:
		tmp = t_1
	elif (z - 1.0) <= 5e+280:
		tmp = ((x - 1.0) * math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(-z) * y) - t)
	tmp = 0.0
	if (Float64(z - 1.0) <= -1e+248)
		tmp = t_1;
	elseif (Float64(z - 1.0) <= 5e+280)
		tmp = Float64(Float64(Float64(x - 1.0) * log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-z * y) - t;
	tmp = 0.0;
	if ((z - 1.0) <= -1e+248)
		tmp = t_1;
	elseif ((z - 1.0) <= 5e+280)
		tmp = ((x - 1.0) * log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[((-z) * y), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(z - 1.0), $MachinePrecision], -1e+248], t$95$1, If[LessEqual[N[(z - 1.0), $MachinePrecision], 5e+280], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -1.00000000000000005e248 or 5.0000000000000002e280 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 35.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 99.8

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

   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 z around inf

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

   8. Applied rewrites 89.0%

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

   if -1.00000000000000005e248 < (-.f64 z #s(literal 1 binary64)) < 5.0000000000000002e280

   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 y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 93.9

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

   5. Applied rewrites 93.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq -1 \cdot 10^{+248}:\\ \;\;\;\;\left(-z\right) \cdot y - t\\ \mathbf{elif}\;z - 1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.2:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= (- x 1.0) -2e+16)
     t_1
     (if (<= (- x 1.0) -0.2) (- (- (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if ((x - 1.0) <= -2e+16) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.2) {
		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 = (x * log(y)) - t
    if ((x - 1.0d0) <= (-2d+16)) then
        tmp = t_1
    else if ((x - 1.0d0) <= (-0.2d0)) 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 = (x * Math.log(y)) - t;
	double tmp;
	if ((x - 1.0) <= -2e+16) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.2) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if (x - 1.0) <= -2e+16:
		tmp = t_1
	elif (x - 1.0) <= -0.2:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -2e+16)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.2)
		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 = (x * log(y)) - t;
	tmp = 0.0;
	if ((x - 1.0) <= -2e+16)
		tmp = t_1;
	elseif ((x - 1.0) <= -0.2)
		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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -2e+16], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.2], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e16 or -0.20000000000000001 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. 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 92.3

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

   5. Applied rewrites 92.3%

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

   if -2e16 < (-.f64 x #s(literal 1 binary64)) < -0.20000000000000001

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x - 1 \leq -0.2:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x - 1 \leq -4 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq 10^{+88}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (- x 1.0) -4e+91)
     t_1
     (if (<= (- x 1.0) 1e+88) (- (* (- 1.0 z) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((x - 1.0) <= -4e+91) {
		tmp = t_1;
	} else if ((x - 1.0) <= 1e+88) {
		tmp = ((1.0 - z) * 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 = x * log(y)
    if ((x - 1.0d0) <= (-4d+91)) then
        tmp = t_1
    else if ((x - 1.0d0) <= 1d+88) then
        tmp = ((1.0d0 - z) * 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 = x * Math.log(y);
	double tmp;
	if ((x - 1.0) <= -4e+91) {
		tmp = t_1;
	} else if ((x - 1.0) <= 1e+88) {
		tmp = ((1.0 - z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (x - 1.0) <= -4e+91:
		tmp = t_1
	elif (x - 1.0) <= 1e+88:
		tmp = ((1.0 - z) * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(x - 1.0) <= -4e+91)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= 1e+88)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((x - 1.0) <= -4e+91)
		tmp = t_1;
	elseif ((x - 1.0) <= 1e+88)
		tmp = ((1.0 - z) * y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -4e+91], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], 1e+88], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4.00000000000000032e91 or 9.99999999999999959e87 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 98.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   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 81.0

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

   7. Applied rewrites 81.0%

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

   if -4.00000000000000032e91 < (-.f64 x #s(literal 1 binary64)) < 9.99999999999999959e87

   1. Initial program 84.1%

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

   3. Taylor expanded in y around 0

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

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

   8. Applied rewrites 63.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -4 \cdot 10^{+91}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x - 1 \leq 10^{+88}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 90.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 99.4

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

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 99.4%

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

Alternative 11: 42.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+25}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+26}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.05e+25) (- t) (if (<= t 2.85e+26) (* (- 1.0 z) y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.05e+25) {
		tmp = -t;
	} else if (t <= 2.85e+26) {
		tmp = (1.0 - z) * y;
	} else {
		tmp = -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 (t <= (-1.05d+25)) then
        tmp = -t
    else if (t <= 2.85d+26) then
        tmp = (1.0d0 - z) * y
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.05e+25) {
		tmp = -t;
	} else if (t <= 2.85e+26) {
		tmp = (1.0 - z) * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.05e+25:
		tmp = -t
	elif t <= 2.85e+26:
		tmp = (1.0 - z) * y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.05e+25)
		tmp = Float64(-t);
	elseif (t <= 2.85e+26)
		tmp = Float64(Float64(1.0 - z) * y);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.05e+25)
		tmp = -t;
	elseif (t <= 2.85e+26)
		tmp = (1.0 - z) * y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.05e+25], (-t), If[LessEqual[t, 2.85e+26], N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e25 or 2.8500000000000002e26 < t

   1. Initial program 96.5%

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

   3. Taylor expanded in 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 75.7

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

   5. Applied rewrites 75.7%

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

   if -1.05e25 < t < 2.8500000000000002e26

   1. Initial program 85.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

   7. Applied rewrites 69.3%

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

   8. Taylor expanded in y around -inf

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

   10. Applied rewrites 16.6%

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

Alternative 12: 46.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 99.4

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

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 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-*.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 44.3%

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

Alternative 13: 46.0% 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 90.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 99.4

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

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 44.2%

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

Alternative 14: 45.8% accurate, 20.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (-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 = (-z * y) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 99.4

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

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 44.1%

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

9. Final simplification 44.1%

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

Alternative 15: 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 90.4%

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

3. Taylor expanded in 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 35.0

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

5. Applied rewrites 35.0%

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

Reproduce

herbie shell --seed 2024296 
(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))