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

Percentage Accurate: 89.3% → 99.8%

Time: 13.5s

Alternatives: 16

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: 89.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. unpow1 N/A

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

   9. unpow1 N/A

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

   10. lift--.f64 N/A

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

   11. *-lft-identity N/A

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

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

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

   13. distribute-lft-neg-in N/A

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

   14. *-lft-identity N/A

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

   15. lower-log1p.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower--.f64 99.9

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

5. Applied rewrites 99.7%

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

Alternative 3: 99.4% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (fma (+ -1.0 x) (log y) (* (* (- z 1.0) 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 - 1.0) * 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(Float64(z - 1.0) * 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[(N[(z - 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(-0.5 * y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
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 99.6%

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

Alternative 4: 94.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+37} \lor \neg \left(x \leq 6.5 \cdot 10^{+15}\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - 1, -y, \left(-\log y\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e+37) (not (<= x 6.5e+15)))
   (fma (+ -1.0 x) (log y) (- t))
   (fma (- z 1.0) (- y) (- (- (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e+37) || !(x <= 6.5e+15)) {
		tmp = fma((-1.0 + x), log(y), -t);
	} else {
		tmp = fma((z - 1.0), -y, (-log(y) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e+37) || !(x <= 6.5e+15))
		tmp = fma(Float64(-1.0 + x), log(y), Float64(-t));
	else
		tmp = fma(Float64(z - 1.0), Float64(-y), Float64(Float64(-log(y)) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e+37], N[Not[LessEqual[x, 6.5e+15]], $MachinePrecision]], N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(z - 1.0), $MachinePrecision] * (-y) + N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999991e37 or 6.5e15 < x

   1. Initial program 95.5%

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

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

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

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 94.8%

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

   if -1.99999999999999991e37 < x < 6.5e15

   1. Initial program 80.8%

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

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. unpow1 N/A

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

      9. unpow1 N/A

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

      10. lift--.f64 N/A

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

      11. *-lft-identity N/A

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

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

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

      13. distribute-lft-neg-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower--.f64 100.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in x around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 96.3

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

   10. Applied rewrites 96.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+37} \lor \neg \left(x \leq 6.5 \cdot 10^{+15}\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - 1, -y, \left(-\log y\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+37} \lor \neg \left(x \leq 6.5 \cdot 10^{+15}\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e+37) (not (<= x 6.5e+15)))
   (fma (+ -1.0 x) (log y) (- t))
   (- (fma (- 1.0 z) y (- (log y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e+37) || !(x <= 6.5e+15)) {
		tmp = fma((-1.0 + x), log(y), -t);
	} else {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e+37) || !(x <= 6.5e+15))
		tmp = fma(Float64(-1.0 + x), log(y), Float64(-t));
	else
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e+37], N[Not[LessEqual[x, 6.5e+15]], $MachinePrecision]], N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999991e37 or 6.5e15 < x

   1. Initial program 95.5%

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

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

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

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 94.8%

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

   if -1.99999999999999991e37 < x < 6.5e15

   1. Initial program 80.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+37} \lor \neg \left(x \leq 6.5 \cdot 10^{+15}\right):\\ \;\;\;\;\mathsf{fma}\left(-1 + x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+35} \lor \neg \left(x - 1 \leq 10^{+14}\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -1e+35) (not (<= (- x 1.0) 1e+14)))
   (- (* (log y) x) t)
   (-
    (*
     (fma (fma (* z (fma -0.25 y -0.3333333333333333)) y (* -0.5 z)) y (- z))
     y)
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -1e+35) || !((x - 1.0) <= 1e+14)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = (fma(fma((z * fma(-0.25, y, -0.3333333333333333)), y, (-0.5 * z)), y, -z) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -1e+35) || !(Float64(x - 1.0) <= 1e+14))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(Float64(fma(fma(Float64(z * fma(-0.25, y, -0.3333333333333333)), y, Float64(-0.5 * z)), y, Float64(-z)) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -1e+35], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], 1e+14]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(z * N[(-0.25 * y + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + (-z)), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -9.9999999999999997e34 or 1e14 < (-.f64 x #s(literal 1 binary64))

   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. *-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 94.0

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

   5. Applied rewrites 94.0%

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

   if -9.9999999999999997e34 < (-.f64 x #s(literal 1 binary64)) < 1e14

   1. Initial program 81.2%

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

   3. Taylor expanded in 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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+35} \lor \neg \left(x - 1 \leq 10^{+14}\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot y - t\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+237}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5e+144)
   (-
    (*
     (fma (fma (* z (fma -0.25 y -0.3333333333333333)) y (* -0.5 z)) y (- z))
     y)
    t)
   (if (<= z 9e+237) (fma (+ -1.0 x) (log y) (- t)) (- (* (- 1.0 z) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5e+144) {
		tmp = (fma(fma((z * fma(-0.25, y, -0.3333333333333333)), y, (-0.5 * z)), y, -z) * y) - t;
	} else if (z <= 9e+237) {
		tmp = fma((-1.0 + x), log(y), -t);
	} else {
		tmp = ((1.0 - z) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5e+144)
		tmp = Float64(Float64(fma(fma(Float64(z * fma(-0.25, y, -0.3333333333333333)), y, Float64(-0.5 * z)), y, Float64(-z)) * y) - t);
	elseif (z <= 9e+237)
		tmp = fma(Float64(-1.0 + x), log(y), Float64(-t));
	else
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5e+144], N[(N[(N[(N[(N[(z * N[(-0.25 * y + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + (-z)), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 9e+237], N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.9999999999999999e144

   1. Initial program 61.0%

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

   3. Taylor expanded in 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 45.9

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.1%

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

   if -4.9999999999999999e144 < z < 8.99999999999999928e237

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

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

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

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

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

      8. remove-double-neg N/A

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

      9. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 95.1%

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

   if 8.99999999999999928e237 < z

   1. Initial program 22.7%

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

   3. Taylor expanded in y around 0

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

      1. +-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 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 16.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 93.2%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot y - t\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+237}:\\ \;\;\;\;\mathsf{fma}\left(-1 + x, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% 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 86.9%

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

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

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

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

   9. log-rec N/A

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

   10. mul-1-neg N/A

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

   11. mul-1-neg N/A

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

   12. log-rec N/A

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

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

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

   14. remove-double-neg N/A

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

   15. *-rgt-identity N/A

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

   16. 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 99.1%

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

6. Final simplification 99.1%

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

Alternative 9: 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 86.9%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 99.1%

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

Alternative 10: 66.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+39} \lor \neg \left(x \leq 1.6 \cdot 10^{+16}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.1e+39) (not (<= x 1.6e+16)))
   (* (log y) x)
   (-
    (*
     (fma (fma (* z (fma -0.25 y -0.3333333333333333)) y (* -0.5 z)) y (- z))
     y)
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e+39) || !(x <= 1.6e+16)) {
		tmp = log(y) * x;
	} else {
		tmp = (fma(fma((z * fma(-0.25, y, -0.3333333333333333)), y, (-0.5 * z)), y, -z) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.1e+39) || !(x <= 1.6e+16))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(fma(fma(Float64(z * fma(-0.25, y, -0.3333333333333333)), y, Float64(-0.5 * z)), y, Float64(-z)) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.1e+39], N[Not[LessEqual[x, 1.6e+16]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(z * N[(-0.25 * y + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + (-z)), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1000000000000003e39 or 1.6e16 < x

   1. Initial program 95.5%

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

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. unpow1 N/A

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

      9. unpow1 N/A

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

      10. lift--.f64 N/A

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

      11. *-lft-identity N/A

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

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

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

      13. distribute-lft-neg-in N/A

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

      14. *-lft-identity N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower--.f64 99.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 74.0

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

   7. Applied rewrites 74.0%

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

   if -3.1000000000000003e39 < x < 1.6e16

   1. Initial program 80.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+39} \lor \neg \left(x \leq 1.6 \cdot 10^{+16}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.5% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
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 99.6%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 35.5%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 52.9%

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

Alternative 12: 46.3% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in 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 40.4

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

5. Applied rewrites 40.4%

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

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 52.7%

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

10. Applied rewrites 52.7%

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

Alternative 13: 46.3% 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 86.9%

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

3. Taylor expanded in 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 40.4

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

5. Applied rewrites 40.4%

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

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 52.7%

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

Alternative 14: 46.3% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
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 99.6%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 35.5%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 52.4%

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

Alternative 15: 46.1% 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 86.9%

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

3. Taylor expanded in 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 40.4

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

5. Applied rewrites 40.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 52.2%

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

Alternative 16: 35.8% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in 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 39.3

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

5. Applied rewrites 39.3%

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

6. Final simplification 39.3%

   \[\leadsto -t \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024342 
(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))