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

Percentage Accurate: 89.0% → 99.3%

Time: 13.9s

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: 89.0% 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: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in 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. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. distribute-lft-neg-out N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 99.5%

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

Alternative 2: 92.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\\ \mathbf{if}\;t\_1 \leq -100000 \lor \neg \left(t\_1 \leq 1000000000\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z - 1, y, \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t)))
   (if (or (<= t_1 -100000.0) (not (<= t_1 1000000000.0)))
     (- (* (+ -1.0 x) (log y)) t)
     (- (fma (- z 1.0) y (log y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
	double tmp;
	if ((t_1 <= -100000.0) || !(t_1 <= 1000000000.0)) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = -fma((z - 1.0), y, log(y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
	tmp = 0.0
	if ((t_1 <= -100000.0) || !(t_1 <= 1000000000.0))
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(-fma(Float64(z - 1.0), y, log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = 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]}, If[Or[LessEqual[t$95$1, -100000.0], N[Not[LessEqual[t$95$1, 1000000000.0]], $MachinePrecision]], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.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)))) t) < -1e5 or 1e9 < (-.f64 (+.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)))) t)

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 92.5%

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

   if -1e5 < (-.f64 (+.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)))) t) < 1e9

   1. Initial program 74.5%

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

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 96.5%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \leq -100000 \lor \neg \left(\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \leq 1000000000\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z - 1, y, \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+20} \lor \neg \left(t\_1 \leq 1000000000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z - 1, y, \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t)))
   (if (or (<= t_1 -2e+20) (not (<= t_1 1000000000.0)))
     (- (* (log y) x) t)
     (- (fma (- z 1.0) y (log y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
	double tmp;
	if ((t_1 <= -2e+20) || !(t_1 <= 1000000000.0)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = -fma((z - 1.0), y, log(y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
	tmp = 0.0
	if ((t_1 <= -2e+20) || !(t_1 <= 1000000000.0))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(-fma(Float64(z - 1.0), y, log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = 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]}, If[Or[LessEqual[t$95$1, -2e+20], N[Not[LessEqual[t$95$1, 1000000000.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], (-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.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)))) t) < -2e20 or 1e9 < (-.f64 (+.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)))) t)

   1. Initial program 94.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if -2e20 < (-.f64 (+.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)))) t) < 1e9

   1. Initial program 73.0%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 93.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \leq -2 \cdot 10^{+20} \lor \neg \left(\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \leq 1000000000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(z - 1, y, \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-7} \lor \neg \left(x \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z - 1, -\left(\log y + t\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.8e-7) (not (<= x 2e-15)))
   (- (* (+ -1.0 x) (log y)) t)
   (fma (- y) (- z 1.0) (- (+ (log y) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.8e-7) || !(x <= 2e-15)) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = fma(-y, (z - 1.0), -(log(y) + t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.8e-7) || !(x <= 2e-15))
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = fma(Float64(-y), Float64(z - 1.0), Float64(-Float64(log(y) + t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.8e-7], N[Not[LessEqual[x, 2e-15]], $MachinePrecision]], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-y) * N[(z - 1.0), $MachinePrecision] + (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000019e-7 or 2.0000000000000002e-15 < x

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 92.4%

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

   if -2.80000000000000019e-7 < x < 2.0000000000000002e-15

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-7} \lor \neg \left(x \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z - 1, -\left(\log y + t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -20000000000000 \lor \neg \left(x - 1 \leq -0.5\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -20000000000000.0) (not (<= (- x 1.0) -0.5)))
   (- (* (log y) x) t)
   (- (+ (log y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -20000000000000.0) || !((x - 1.0) <= -0.5)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = -(log(y) + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x - 1.0d0) <= (-20000000000000.0d0)) .or. (.not. ((x - 1.0d0) <= (-0.5d0)))) then
        tmp = (log(y) * x) - t
    else
        tmp = -(log(y) + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -20000000000000.0) || !((x - 1.0) <= -0.5)) {
		tmp = (Math.log(y) * x) - t;
	} else {
		tmp = -(Math.log(y) + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - 1.0) <= -20000000000000.0) or not ((x - 1.0) <= -0.5):
		tmp = (math.log(y) * x) - t
	else:
		tmp = -(math.log(y) + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -20000000000000.0) || !(Float64(x - 1.0) <= -0.5))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(-Float64(log(y) + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x - 1.0) <= -20000000000000.0) || ~(((x - 1.0) <= -0.5)))
		tmp = (log(y) * x) - t;
	else
		tmp = -(log(y) + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -20000000000000.0], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], -0.5]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e13 or -0.5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -2e13 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 82.7%

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

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 81.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -20000000000000 \lor \neg \left(x - 1 \leq -0.5\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-7} \lor \neg \left(x \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \mathsf{fma}\left(y, z, \log y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.8e-7) (not (<= x 2e-15)))
   (- (* (+ -1.0 x) (log y)) t)
   (- (- y (fma y z (log y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.8e-7) || !(x <= 2e-15)) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = (y - fma(y, z, log(y))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.8e-7) || !(x <= 2e-15))
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(Float64(y - fma(y, z, log(y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.8e-7], N[Not[LessEqual[x, 2e-15]], $MachinePrecision]], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[(y * z + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.80000000000000019e-7 or 2.0000000000000002e-15 < x

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 92.4%

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

   if -2.80000000000000019e-7 < x < 2.0000000000000002e-15

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 99.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-7} \lor \neg \left(x \leq 2 \cdot 10^{-15}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \mathsf{fma}\left(y, z, \log y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+47} \lor \neg \left(x - 1 \leq 2 \cdot 10^{+74}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -1e+47) (not (<= (- x 1.0) 2e+74)))
   (* (log y) x)
   (- (+ (log y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -1e+47) || !((x - 1.0) <= 2e+74)) {
		tmp = log(y) * x;
	} else {
		tmp = -(log(y) + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x - 1.0d0) <= (-1d+47)) .or. (.not. ((x - 1.0d0) <= 2d+74))) then
        tmp = log(y) * x
    else
        tmp = -(log(y) + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -1e+47) || !((x - 1.0) <= 2e+74)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = -(Math.log(y) + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - 1.0) <= -1e+47) or not ((x - 1.0) <= 2e+74):
		tmp = math.log(y) * x
	else:
		tmp = -(math.log(y) + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -1e+47) || !(Float64(x - 1.0) <= 2e+74))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(-Float64(log(y) + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x - 1.0) <= -1e+47) || ~(((x - 1.0) <= 2e+74)))
		tmp = log(y) * x;
	else
		tmp = -(log(y) + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -1e+47], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], 2e+74]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e47 or 1.9999999999999999e74 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. distribute-lft-neg-out N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 74.8

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

   8. Applied rewrites 74.8%

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

   if -1e47 < (-.f64 x #s(literal 1 binary64)) < 1.9999999999999999e74

   1. Initial program 84.2%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 78.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+47} \lor \neg \left(x - 1 \leq 2 \cdot 10^{+74}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq -4 \cdot 10^{+142}:\\ \;\;\;\;\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \mathbf{elif}\;z - 1 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) -4e+142)
   (-
    (* (* (- (* (- (* (- (* -0.25 y) 0.3333333333333333) y) 0.5) y) 1.0) y) z)
    t)
   (if (<= (- z 1.0) 2e+127)
     (- (+ (log y) t))
     (- (* (* (- (* -0.5 y) 1.0) y) z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= -4e+142) {
		tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t;
	} else if ((z - 1.0) <= 2e+127) {
		tmp = -(log(y) + t);
	} else {
		tmp = ((((-0.5 * y) - 1.0) * y) * z) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z - 1.0d0) <= (-4d+142)) then
        tmp = (((((((((-0.25d0) * y) - 0.3333333333333333d0) * y) - 0.5d0) * y) - 1.0d0) * y) * z) - t
    else if ((z - 1.0d0) <= 2d+127) then
        tmp = -(log(y) + t)
    else
        tmp = (((((-0.5d0) * y) - 1.0d0) * y) * z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= -4e+142) {
		tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t;
	} else if ((z - 1.0) <= 2e+127) {
		tmp = -(Math.log(y) + t);
	} else {
		tmp = ((((-0.5 * y) - 1.0) * y) * z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z - 1.0) <= -4e+142:
		tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t
	elif (z - 1.0) <= 2e+127:
		tmp = -(math.log(y) + t)
	else:
		tmp = ((((-0.5 * y) - 1.0) * y) * z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= -4e+142)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t);
	elseif (Float64(z - 1.0) <= 2e+127)
		tmp = Float64(-Float64(log(y) + t));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * y) * z) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z - 1.0) <= -4e+142)
		tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t;
	elseif ((z - 1.0) <= 2e+127)
		tmp = -(log(y) + t);
	else
		tmp = ((((-0.5 * y) - 1.0) * y) * z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], -4e+142], N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.25 * y), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(z - 1.0), $MachinePrecision], 2e+127], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -4.0000000000000002e142

   1. Initial program 63.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 27.9

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.4%

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

   if -4.0000000000000002e142 < (-.f64 z #s(literal 1 binary64)) < 1.99999999999999991e127

   1. Initial program 98.6%

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

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 65.8%

       \[\leadsto \left(-\log y\right) - t \]

   if 1.99999999999999991e127 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 57.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 36.6

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.0%

      \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq -4 \cdot 10^{+142}:\\ \;\;\;\;\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \mathbf{elif}\;z - 1 \leq 2 \cdot 10^{+127}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in 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. associate--l+ N/A

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

   2. associate-*r* N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-rgt-identity N/A

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

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

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

5. Applied rewrites 99.5%

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

Alternative 10: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in 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. associate--l+ N/A

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

   2. associate-*r* N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-rgt-identity N/A

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

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

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 99.5%

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

Alternative 11: 47.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((((((((-0.25d0) * y) - 0.3333333333333333d0) * y) - 0.5d0) * y) - 1.0d0) * y) * z) - t
end function

Java

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

Python

def code(x, y, z, t):
	return ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t;
end

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 36.2

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

5. Applied rewrites 36.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 47.4%

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

Alternative 12: 47.2% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((((-0.5d0) * y) - 1.0d0) * y) * z) - t
end function

Java

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

Python

def code(x, y, z, t):
	return ((((-0.5 * y) - 1.0) * y) * z) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 36.2

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

5. Applied rewrites 36.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 47.4%

   \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot y\right) \cdot z - t \]
9. Add Preprocessing

Alternative 13: 44.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -940000000 \lor \neg \left(t \leq 78000000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -940000000.0) (not (<= t 78000000000000.0)))
   (- t)
   (* (- 1.0 z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -940000000.0) || !(t <= 78000000000000.0)) {
		tmp = -t;
	} else {
		tmp = (1.0 - z) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-940000000.0d0)) .or. (.not. (t <= 78000000000000.0d0))) then
        tmp = -t
    else
        tmp = (1.0d0 - z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -940000000.0) || !(t <= 78000000000000.0)) {
		tmp = -t;
	} else {
		tmp = (1.0 - z) * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -940000000.0) or not (t <= 78000000000000.0):
		tmp = -t
	else:
		tmp = (1.0 - z) * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -940000000.0) || !(t <= 78000000000000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(1.0 - z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -940000000.0) || ~((t <= 78000000000000.0)))
		tmp = -t;
	else
		tmp = (1.0 - z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -940000000.0], N[Not[LessEqual[t, 78000000000000.0]], $MachinePrecision]], (-t), N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.4e8 or 7.8e13 < t

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -9.4e8 < t < 7.8e13

   1. Initial program 78.9%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 23.3%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -940000000 \lor \neg \left(t \leq 78000000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -940000000 \lor \neg \left(t \leq 78000000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -940000000.0) (not (<= t 78000000000000.0))) (- t) (* (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -940000000.0) || !(t <= 78000000000000.0)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-940000000.0d0)) .or. (.not. (t <= 78000000000000.0d0))) then
        tmp = -t
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -940000000.0) || !(t <= 78000000000000.0)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -940000000.0) or not (t <= 78000000000000.0):
		tmp = -t
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -940000000.0) || !(t <= 78000000000000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -940000000.0) || ~((t <= 78000000000000.0)))
		tmp = -t;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -940000000.0], N[Not[LessEqual[t, 78000000000000.0]], $MachinePrecision]], (-t), N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.4e8 or 7.8e13 < t

   1. Initial program 97.4%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -9.4e8 < t < 7.8e13

   1. Initial program 78.9%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 22.9%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -940000000 \lor \neg \left(t \leq 78000000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 36.2

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

5. Applied rewrites 36.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 47.4%

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

Alternative 16: 47.0% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 36.2

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

5. Applied rewrites 36.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 47.4%

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

Alternative 17: 36.5% 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 87.3%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.6

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

5. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024338 
(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))