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

Percentage Accurate: 89.1% → 99.8%

Time: 16.4s

Alternatives: 17

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.1% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\\ \mathbf{if}\;t\_2 \leq -200000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 170:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(-y\right) \cdot y\right) \cdot y}{\mathsf{fma}\left(y, y, 0 \cdot y\right)}, z, y\right) - t\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(-1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t))
        (t_2 (+ (* (log (- 1.0 y)) (- z 1.0)) (* (log y) (- x 1.0)))))
   (if (<= t_2 -200000000000.0)
     t_1
     (if (<= t_2 170.0)
       (- (fma (/ (* (* (- y) y) y) (fma y y (* 0.0 y))) z y) t)
       (if (<= t_2 5e+16) (fma -1.0 (log y) (- t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0));
	double tmp;
	if (t_2 <= -200000000000.0) {
		tmp = t_1;
	} else if (t_2 <= 170.0) {
		tmp = fma((((-y * y) * y) / fma(y, y, (0.0 * y))), z, y) - t;
	} else if (t_2 <= 5e+16) {
		tmp = fma(-1.0, log(y), -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(log(y) * Float64(x - 1.0)))
	tmp = 0.0
	if (t_2 <= -200000000000.0)
		tmp = t_1;
	elseif (t_2 <= 170.0)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(-y) * y) * y) / fma(y, y, Float64(0.0 * y))), z, y) - t);
	elseif (t_2 <= 5e+16)
		tmp = fma(-1.0, log(y), Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -200000000000.0], t$95$1, If[LessEqual[t$95$2, 170.0], N[(N[(N[(N[(N[((-y) * y), $MachinePrecision] * y), $MachinePrecision] / N[(y * y + N[(0.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 5e+16], N[(-1.0 * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -2e11 or 5e16 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if -2e11 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 170

   1. Initial program 72.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   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 86.9%

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

   10. Applied rewrites 86.9%

       \[\leadsto \mathsf{fma}\left(\frac{0 - \left(y \cdot y\right) \cdot y}{0 + \mathsf{fma}\left(y, y, 0 \cdot y\right)}, z, y\right) - t \]

   if 170 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 5e16

   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. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 84.5%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right) \leq -200000000000:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right) \leq 170:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(\left(-y\right) \cdot y\right) \cdot y}{\mathsf{fma}\left(y, y, 0 \cdot y\right)}, z, y\right) - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right) \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(-1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - z, y, \log y \cdot x\right) - t\\ \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \log y, \mathsf{fma}\left(-y, z, y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma (- 1.0 z) y (* (log y) x)) t)))
   (if (<= (- x 1.0) -5e+24)
     t_1
     (if (<= (- x 1.0) 2000000.0)
       (fma -1.0 (log y) (- (fma (- y) z y) t))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((1.0 - z), y, (log(y) * x)) - t;
	double tmp;
	if ((x - 1.0) <= -5e+24) {
		tmp = t_1;
	} else if ((x - 1.0) <= 2000000.0) {
		tmp = fma(-1.0, log(y), (fma(-y, z, y) - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(1.0 - z), y, Float64(log(y) * x)) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -5e+24)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= 2000000.0)
		tmp = fma(-1.0, log(y), Float64(fma(Float64(-y), z, y) - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(1.0 - z), $MachinePrecision] * y + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+24], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], 2000000.0], N[(-1.0 * N[Log[y], $MachinePrecision] + N[(N[((-y) * z + y), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5.00000000000000045e24 or 2e6 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.4%

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

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

   8. Applied rewrites 98.9%

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

   if -5.00000000000000045e24 < (-.f64 x #s(literal 1 binary64)) < 2e6

   1. Initial program 81.7%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing
   3. 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. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

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

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

      11. mul-1-neg N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot x\right) - t\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \log y, \mathsf{fma}\left(-y, z, y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - z, y, \log y \cdot x\right) - t\\ \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma (- 1.0 z) y (* (log y) x)) t)))
   (if (<= (- x 1.0) -5e+24)
     t_1
     (if (<= (- x 1.0) 2000000.0) (- (fma (- 1.0 z) y (- (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((1.0 - z), y, (log(y) * x)) - t;
	double tmp;
	if ((x - 1.0) <= -5e+24) {
		tmp = t_1;
	} else if ((x - 1.0) <= 2000000.0) {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(1.0 - z), y, Float64(log(y) * x)) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -5e+24)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= 2000000.0)
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(1.0 - z), $MachinePrecision] * y + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+24], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], 2000000.0], N[(N[(N[(1.0 - z), $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)) < -5.00000000000000045e24 or 2e6 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.4%

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

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

   8. Applied rewrites 98.9%

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

   if -5.00000000000000045e24 < (-.f64 x #s(literal 1 binary64)) < 2e6

   1. Initial program 81.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot x\right) - t\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x 1.0) -5e+24)
   (- (* (log y) x) t)
   (if (<= (- x 1.0) 2000000.0)
     (- (fma (- 1.0 z) y (- (log y))) t)
     (fma (- x 1.0) (log y) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - 1.0) <= -5e+24) {
		tmp = (log(y) * x) - t;
	} else if ((x - 1.0) <= 2000000.0) {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	} else {
		tmp = fma((x - 1.0), log(y), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -5e+24)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (Float64(x - 1.0) <= 2000000.0)
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+24], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], 2000000.0], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5.00000000000000045e24

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 95.0

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

   5. Applied rewrites 95.0%

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

   if -5.00000000000000045e24 < (-.f64 x #s(literal 1 binary64)) < 2e6

   1. Initial program 81.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   if 2e6 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.5%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(-z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x 1.0) -5e+24)
   (- (* (log y) x) t)
   (if (<= (- x 1.0) 2000000.0)
     (- (fma (- z) y (- (log y))) t)
     (fma (- x 1.0) (log y) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - 1.0) <= -5e+24) {
		tmp = (log(y) * x) - t;
	} else if ((x - 1.0) <= 2000000.0) {
		tmp = fma(-z, y, -log(y)) - t;
	} else {
		tmp = fma((x - 1.0), log(y), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -5e+24)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (Float64(x - 1.0) <= 2000000.0)
		tmp = Float64(fma(Float64(-z), y, Float64(-log(y))) - t);
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+24], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], 2000000.0], N[(N[((-z) * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5.00000000000000045e24

   1. Initial program 95.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 95.0

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

   5. Applied rewrites 95.0%

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

   if -5.00000000000000045e24 < (-.f64 x #s(literal 1 binary64)) < 2e6

   1. Initial program 81.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 98.6%

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

   if 2e6 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.5%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(-z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower--.f64 N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-neg.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 8: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	return Float64(fma(Float64(Float64(z - 1.0) * y), fma(-0.5, y, -1.0), Float64(log(y) * Float64(x - 1.0))) - 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[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
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 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 9: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= (- x 1.0) -5e+24)
     t_1
     (if (<= (- x 1.0) 2000000.0) (- (fma (- y) z y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if ((x - 1.0) <= -5e+24) {
		tmp = t_1;
	} else if ((x - 1.0) <= 2000000.0) {
		tmp = fma(-y, z, y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -5e+24)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= 2000000.0)
		tmp = Float64(fma(Float64(-y), z, y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+24], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], 2000000.0], N[(N[((-y) * z + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5.00000000000000045e24 or 2e6 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -5.00000000000000045e24 < (-.f64 x #s(literal 1 binary64)) < 2e6

   1. Initial program 81.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate--r+ N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 65.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x - 1 \leq 2000000:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= (- x 1.0) -5e+110)
     t_1
     (if (<= (- x 1.0) 5e+97) (- (fma (- y) z y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((x - 1.0) <= -5e+110) {
		tmp = t_1;
	} else if ((x - 1.0) <= 5e+97) {
		tmp = fma(-y, z, y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (Float64(x - 1.0) <= -5e+110)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= 5e+97)
		tmp = Float64(fma(Float64(-y), z, y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+110], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], 5e+97], N[(N[((-y) * z + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4.99999999999999978e110 or 4.99999999999999999e97 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -4.99999999999999978e110 < (-.f64 x #s(literal 1 binary64)) < 4.99999999999999999e97

   1. Initial program 83.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 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 64.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+110}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x - 1 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq -2 \cdot 10^{+241}:\\ \;\;\;\;\left(-z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) -2e+241) (- (* (- z) y) t) (- (fma (- x 1.0) (log y) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= -2e+241) {
		tmp = (-z * y) - t;
	} else {
		tmp = fma((x - 1.0), log(y), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= -2e+241)
		tmp = Float64(Float64(Float64(-z) * y) - t);
	else
		tmp = Float64(fma(Float64(x - 1.0), log(y), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], -2e+241], N[(N[((-z) * y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -2.0000000000000001e241

   1. Initial program 37.8%

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

   3. Taylor expanded in 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 84.2%

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

   if -2.0000000000000001e241 < (-.f64 z #s(literal 1 binary64))

   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 0

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

   8. Applied rewrites 89.4%

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

Alternative 12: 88.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq -2 \cdot 10^{+241}:\\ \;\;\;\;\left(-z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) -2e+241) (- (* (- z) y) t) (fma (- x 1.0) (log y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= -2e+241) {
		tmp = (-z * y) - t;
	} else {
		tmp = fma((x - 1.0), log(y), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= -2e+241)
		tmp = Float64(Float64(Float64(-z) * y) - t);
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], -2e+241], N[(N[((-z) * y), $MachinePrecision] - t), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -2.0000000000000001e241

   1. Initial program 37.8%

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

   3. Taylor expanded in 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 84.2%

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

   if -2.0000000000000001e241 < (-.f64 z #s(literal 1 binary64))

   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

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 89.4

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

   5. Applied rewrites 89.4%

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

Alternative 13: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. mul-1-neg N/A

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-in N/A

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

   9. *-rgt-identity N/A

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

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

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

   11. mul-1-neg N/A

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

   12. associate-*r* N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 14: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 15: 46.1% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 49.6%

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

Alternative 16: 45.9% 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 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 49.5%

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

Alternative 17: 35.5% 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 88.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 39.2

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

5. Applied rewrites 39.2%

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

Reproduce

herbie shell --seed 2024235 
(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))