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

Percentage Accurate: 89.3% → 99.5%

Time: 13.7s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% 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: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 93.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t\\ \mathbf{if}\;t\_1 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{elif}\;t\_1 \leq 600:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (+ (* (log (- 1.0 y)) (- z 1.0)) (* (log y) (- x 1.0))) t)))
   (if (<= t_1 20.0)
     (- (fma (- x 1.0) (log y) y) t)
     (if (<= t_1 600.0)
       (fma (- 1.0 z) y (- (log y)))
       (fma (- x 1.0) (log y) (- t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0))) - t;
	double tmp;
	if (t_1 <= 20.0) {
		tmp = fma((x - 1.0), log(y), y) - t;
	} else if (t_1 <= 600.0) {
		tmp = fma((1.0 - z), y, -log(y));
	} else {
		tmp = fma((x - 1.0), log(y), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(log(y) * Float64(x - 1.0))) - t)
	tmp = 0.0
	if (t_1 <= 20.0)
		tmp = Float64(fma(Float64(x - 1.0), log(y), y) - t);
	elseif (t_1 <= 600.0)
		tmp = fma(Float64(1.0 - z), y, Float64(-log(y)));
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, 20.0], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 600.0], N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]]]

Derivation

1. Split input into 3 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) < 20

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.1%

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

   if 20 < (-.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) < 600

   1. Initial program 64.9%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.3%

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

   if 600 < (-.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 96.2%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t \leq 20:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{elif}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t \leq 600:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - 1, \log y, -t\right)\\ t_2 := \left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 600:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- x 1.0) (log y) (- t)))
        (t_2 (- (+ (* (log (- 1.0 y)) (- z 1.0)) (* (log y) (- x 1.0))) t)))
   (if (<= t_2 -5e+15)
     t_1
     (if (<= t_2 600.0) (fma (- 1.0 z) y (- (log y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((x - 1.0), log(y), -t);
	double t_2 = ((log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0))) - t;
	double tmp;
	if (t_2 <= -5e+15) {
		tmp = t_1;
	} else if (t_2 <= 600.0) {
		tmp = fma((1.0 - z), y, -log(y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(x - 1.0), log(y), Float64(-t))
	t_2 = Float64(Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(log(y) * Float64(x - 1.0))) - t)
	tmp = 0.0
	if (t_2 <= -5e+15)
		tmp = t_1;
	elseif (t_2 <= 600.0)
		tmp = fma(Float64(1.0 - z), y, Float64(-log(y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+15], t$95$1, If[LessEqual[t$95$2, 600.0], N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision], t$95$1]]]]

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) < -5e15 or 600 < (-.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 95.3%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 95.0

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

   5. Applied rewrites 95.0%

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

   if -5e15 < (-.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) < 600

   1. Initial program 65.6%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t \leq -5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{elif}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t \leq 600:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ t_2 := \left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t))
        (t_2 (- (+ (* (log (- 1.0 y)) (- z 1.0)) (* (log y) (- x 1.0))) t)))
   (if (<= t_2 -5e+15)
     t_1
     (if (<= t_2 1000.0) (fma (- 1.0 z) y (- (log y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double t_2 = ((log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0))) - t;
	double tmp;
	if (t_2 <= -5e+15) {
		tmp = t_1;
	} else if (t_2 <= 1000.0) {
		tmp = fma((1.0 - z), y, -log(y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	t_2 = Float64(Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(log(y) * Float64(x - 1.0))) - t)
	tmp = 0.0
	if (t_2 <= -5e+15)
		tmp = t_1;
	elseif (t_2 <= 1000.0)
		tmp = fma(Float64(1.0 - z), y, Float64(-log(y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+15], t$95$1, If[LessEqual[t$95$2, 1000.0], N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision], t$95$1]]]]

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) < -5e15 or 1e3 < (-.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 95.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 94.2

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

   5. Applied rewrites 94.2%

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

   if -5e15 < (-.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) < 1e3

   1. Initial program 69.5%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.4%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t \leq -5 \cdot 10^{+15}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{if}\;x - 1 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.9999998:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\left(t + \log y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- x 1.0) (log y) (- t))))
   (if (<= (- x 1.0) -200000.0)
     t_1
     (if (<= (- x 1.0) -0.9999998) (fma (- 1.0 z) y (- (+ t (log y)))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((x - 1.0), log(y), -t);
	double tmp;
	if ((x - 1.0) <= -200000.0) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.9999998) {
		tmp = fma((1.0 - z), y, -(t + log(y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(x - 1.0), log(y), Float64(-t))
	tmp = 0.0
	if (Float64(x - 1.0) <= -200000.0)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.9999998)
		tmp = fma(Float64(1.0 - z), y, Float64(-Float64(t + log(y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -200000.0], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.9999998], N[(N[(1.0 - z), $MachinePrecision] * y + (-N[(t + N[Log[y], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e5 or -0.999999799999999994 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if -2e5 < (-.f64 x #s(literal 1 binary64)) < -0.999999799999999994

   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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.7%

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

Alternative 6: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot \left(x - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.5e-10)
   (- (fma (- x 1.0) (log y) y) t)
   (if (<= t 1.35e-8)
     (fma (- 1.0 z) y (* (log y) (- x 1.0)))
     (fma (- x 1.0) (log y) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.5e-10) {
		tmp = fma((x - 1.0), log(y), y) - t;
	} else if (t <= 1.35e-8) {
		tmp = fma((1.0 - z), y, (log(y) * (x - 1.0)));
	} else {
		tmp = fma((x - 1.0), log(y), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.5e-10)
		tmp = Float64(fma(Float64(x - 1.0), log(y), y) - t);
	elseif (t <= 1.35e-8)
		tmp = fma(Float64(1.0 - z), y, Float64(log(y) * Float64(x - 1.0)));
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.5e-10], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 1.35e-8], N[(N[(1.0 - z), $MachinePrecision] * y + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.49999999999999995e-10

   1. Initial program 98.2%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 98.2%

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

   if -7.49999999999999995e-10 < t < 1.35000000000000001e-8

   1. Initial program 80.5%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 99.6%

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

   if 1.35000000000000001e-8 < t

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot \left(x - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, \log y \cdot \left(x - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.5e-10)
   (- (fma (- x 1.0) (log y) y) t)
   (if (<= t 1.35e-8)
     (fma (- z) y (* (log y) (- x 1.0)))
     (fma (- x 1.0) (log y) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.5e-10) {
		tmp = fma((x - 1.0), log(y), y) - t;
	} else if (t <= 1.35e-8) {
		tmp = fma(-z, y, (log(y) * (x - 1.0)));
	} else {
		tmp = fma((x - 1.0), log(y), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.5e-10)
		tmp = Float64(fma(Float64(x - 1.0), log(y), y) - t);
	elseif (t <= 1.35e-8)
		tmp = fma(Float64(-z), y, Float64(log(y) * Float64(x - 1.0)));
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.5e-10], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 1.35e-8], N[((-z) * y + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.49999999999999995e-10

   1. Initial program 98.2%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 98.2%

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

   if -7.49999999999999995e-10 < t < 1.35000000000000001e-8

   1. Initial program 80.5%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 99.3%

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

   if 1.35000000000000001e-8 < t

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-z, y, \log y \cdot \left(x - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -5500:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 15000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= x -5500.0) t_1 (if (<= x 15000.0) (fma (- 1.0 z) y (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if (x <= -5500.0) {
		tmp = t_1;
	} else if (x <= 15000.0) {
		tmp = fma((1.0 - z), y, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	tmp = 0.0
	if (x <= -5500.0)
		tmp = t_1;
	elseif (x <= 15000.0)
		tmp = fma(Float64(1.0 - z), y, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -5500.0], t$95$1, If[LessEqual[x, 15000.0], N[(N[(1.0 - z), $MachinePrecision] * y + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5500 or 15000 < x

   1. Initial program 94.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if -5500 < x < 15000

   1. Initial program 84.4%

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

   3. Taylor expanded in y around 0

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

      1. 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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 70.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5500:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq 15000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in y around 0

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

   1. 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. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 N/A

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

   18. lower-neg.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 10: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -290000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -290000.0) t_1 (if (<= x 1.06e+69) (fma (- 1.0 z) y (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -290000.0) {
		tmp = t_1;
	} else if (x <= 1.06e+69) {
		tmp = fma((1.0 - z), y, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -290000.0)
		tmp = t_1;
	elseif (x <= 1.06e+69)
		tmp = fma(Float64(1.0 - z), y, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -290000.0], t$95$1, If[LessEqual[x, 1.06e+69], N[(N[(1.0 - z), $MachinePrecision] * y + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e5 or 1.06000000000000004e69 < x

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -2.9e5 < x < 1.06000000000000004e69

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0

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

      1. 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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 68.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -290000:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-10}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e-10) (- t) (if (<= t 1.15e+31) (* (- 1.0 z) y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e-10) {
		tmp = -t;
	} else if (t <= 1.15e+31) {
		tmp = (1.0 - z) * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9d-10)) then
        tmp = -t
    else if (t <= 1.15d+31) then
        tmp = (1.0d0 - z) * y
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e-10) {
		tmp = -t;
	} else if (t <= 1.15e+31) {
		tmp = (1.0 - z) * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e-10:
		tmp = -t
	elif t <= 1.15e+31:
		tmp = (1.0 - z) * y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e-10)
		tmp = Float64(-t);
	elseif (t <= 1.15e+31)
		tmp = Float64(Float64(1.0 - z) * y);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e-10)
		tmp = -t;
	elseif (t <= 1.15e+31)
		tmp = (1.0 - z) * y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e-10], (-t), If[LessEqual[t, 1.15e+31], N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -8.9999999999999999e-10 or 1.15e31 < t

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if -8.9999999999999999e-10 < t < 1.15e31

   1. Initial program 81.4%

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

   3. Taylor expanded in y around 0

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

      1. 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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 20.4%

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

Alternative 12: 42.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-10}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9e-10) (- t) (if (<= t 1.15e+31) (* (- y) z) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e-10) {
		tmp = -t;
	} else if (t <= 1.15e+31) {
		tmp = -y * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9d-10)) then
        tmp = -t
    else if (t <= 1.15d+31) then
        tmp = -y * z
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9e-10) {
		tmp = -t;
	} else if (t <= 1.15e+31) {
		tmp = -y * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9e-10:
		tmp = -t
	elif t <= 1.15e+31:
		tmp = -y * z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9e-10)
		tmp = Float64(-t);
	elseif (t <= 1.15e+31)
		tmp = Float64(Float64(-y) * z);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9e-10)
		tmp = -t;
	elseif (t <= 1.15e+31)
		tmp = -y * z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9e-10], (-t), If[LessEqual[t, 1.15e+31], N[((-y) * z), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -8.9999999999999999e-10 or 1.15e31 < t

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if -8.9999999999999999e-10 < t < 1.15e31

   1. Initial program 81.4%

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

   3. Taylor expanded in y around 0

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

      1. 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. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 19.9%

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

Alternative 13: 45.9% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in y around 0

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

   1. 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. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 N/A

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

   18. lower-neg.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 46.9%

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

Alternative 14: 45.7% 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 89.7%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 46.8

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

5. Applied rewrites 46.8%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 46.8%

   \[\leadsto z \cdot \left(-y\right) - t \]

9. Final simplification 46.8%

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

Alternative 15: 35.4% 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 89.7%

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

3. Taylor expanded in 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 37.1

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

5. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2024243 
(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))