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

Percentage Accurate: 89.2% → 99.8%

Time: 18.3s

Alternatives: 17

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log1p(-y) / (1.0 / (z + -1.0))) + (Math.log(y) * (x + -1.0))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log1p(-y) / (1.0 / (z + -1.0))) + (math.log(y) * (x + -1.0))) - t

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. clear-num N/A

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

   14. flip-- N/A

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

   15. lift--.f64 N/A

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

   16. lower-/.f64 99.7

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

   17. lift--.f64 N/A

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

   18. sub-neg N/A

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

   19. lower-+.f64 N/A

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

   20. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \left(\frac{\mathsf{log1p}\left(-y\right)}{\frac{1}{z + -1}} + \log y \cdot \left(x + -1\right)\right) - t \]
6. Add Preprocessing

Alternative 2: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - z, x \cdot \log y\right) - t\\ t_2 := \log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq -1.5 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma y (- 1.0 z) (* x (log y))) t))
        (t_2 (+ (* (log y) (+ x -1.0)) (* (+ z -1.0) (log (- 1.0 y))))))
   (if (<= t_2 -1.5e+14)
     t_1
     (if (<= t_2 1000.0) (- (fma y (- 1.0 z) (- (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (1.0 - z), (x * log(y))) - t;
	double t_2 = (log(y) * (x + -1.0)) + ((z + -1.0) * log((1.0 - y)));
	double tmp;
	if (t_2 <= -1.5e+14) {
		tmp = t_1;
	} else if (t_2 <= 1000.0) {
		tmp = fma(y, (1.0 - z), -log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(y, Float64(1.0 - z), Float64(x * log(y))) - t)
	t_2 = Float64(Float64(log(y) * Float64(x + -1.0)) + Float64(Float64(z + -1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_2 <= -1.5e+14)
		tmp = t_1;
	elseif (t_2 <= 1000.0)
		tmp = Float64(fma(y, Float64(1.0 - z), Float64(-log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * N[(1.0 - z), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.5e+14], t$95$1, If[LessEqual[t$95$2, 1000.0], N[(N[(y * N[(1.0 - z), $MachinePrecision] + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -1.5e14 or 1e3 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 89.9%

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

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.5%

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

   if -1.5e14 < (+.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)))) < 1e3

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right) \leq -1.5 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, x \cdot \log y\right) - t\\ \mathbf{elif}\;\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right) \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, x \cdot \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right)\right) - t\\ t_2 := x \cdot \log y - t\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 20000000000:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (+ (* (log y) (+ x -1.0)) (* (+ z -1.0) (log (- 1.0 y)))) t))
        (t_2 (- (* x (log y)) t)))
   (if (<= t_1 -1e+14) t_2 (if (<= t_1 20000000000.0) (- (- (log y)) t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((log(y) * (x + -1.0)) + ((z + -1.0) * log((1.0 - y)))) - t;
	double t_2 = (x * log(y)) - t;
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_2;
	} else if (t_1 <= 20000000000.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((log(y) * (x + (-1.0d0))) + ((z + (-1.0d0)) * log((1.0d0 - y)))) - t
    t_2 = (x * log(y)) - t
    if (t_1 <= (-1d+14)) then
        tmp = t_2
    else if (t_1 <= 20000000000.0d0) then
        tmp = -log(y) - t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((Math.log(y) * (x + -1.0)) + ((z + -1.0) * Math.log((1.0 - y)))) - t;
	double t_2 = (x * Math.log(y)) - t;
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_2;
	} else if (t_1 <= 20000000000.0) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((math.log(y) * (x + -1.0)) + ((z + -1.0) * math.log((1.0 - y)))) - t
	t_2 = (x * math.log(y)) - t
	tmp = 0
	if t_1 <= -1e+14:
		tmp = t_2
	elif t_1 <= 20000000000.0:
		tmp = -math.log(y) - t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(log(y) * Float64(x + -1.0)) + Float64(Float64(z + -1.0) * log(Float64(1.0 - y)))) - t)
	t_2 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (t_1 <= -1e+14)
		tmp = t_2;
	elseif (t_1 <= 20000000000.0)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((log(y) * (x + -1.0)) + ((z + -1.0) * log((1.0 - y)))) - t;
	t_2 = (x * log(y)) - t;
	tmp = 0.0;
	if (t_1 <= -1e+14)
		tmp = t_2;
	elseif (t_1 <= 20000000000.0)
		tmp = -log(y) - t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+14], t$95$2, If[LessEqual[t$95$1, 20000000000.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$2]]]]

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) < -1e14 or 2e10 < (-.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 90.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if -1e14 < (-.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) < 2e10

   1. Initial program 77.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 77.4

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
   6. 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. mul-1-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 26.2

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 75.3

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

   10. Applied rewrites 75.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 73.1%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right)\right) - t \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\left(\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \log \left(1 - y\right)\right) - t \leq 20000000000:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 87.7

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 87.7

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

4. Applied rewrites 87.7%

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

5. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. lower-fma.f64 N/A

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

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

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

Alternative 6: 94.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -1 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ x -1.0) -1e+62)
   (- (* x (log y)) t)
   (if (<= (+ x -1.0) -1.0)
     (- (fma y (- 1.0 z) (- (log y))) t)
     (- (fma (log y) (+ x -1.0) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x + -1.0) <= -1e+62) {
		tmp = (x * log(y)) - t;
	} else if ((x + -1.0) <= -1.0) {
		tmp = fma(y, (1.0 - z), -log(y)) - t;
	} else {
		tmp = fma(log(y), (x + -1.0), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -1e+62)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(x + -1.0) <= -1.0)
		tmp = Float64(fma(y, Float64(1.0 - z), Float64(-log(y))) - t);
	else
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x + -1.0), $MachinePrecision], -1e+62], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x + -1.0), $MachinePrecision], -1.0], N[(N[(y * N[(1.0 - z), $MachinePrecision] + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.00000000000000004e62

   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 x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if -1.00000000000000004e62 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 83.8%

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

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.7%

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

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

   1. Initial program 87.7%

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

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 87.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -1 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

6. Final simplification 99.0%

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

Alternative 8: 89.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -1 \cdot 10^{+208}:\\ \;\;\;\;y \cdot \left(z \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{elif}\;z + -1 \leq 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -1e+208)
   (- (* y (* z (fma y -0.5 -1.0))) t)
   (if (<= (+ z -1.0) 1e+264)
     (- (fma (log y) (+ x -1.0) y) t)
     (- (* y (- z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -1e+208) {
		tmp = (y * (z * fma(y, -0.5, -1.0))) - t;
	} else if ((z + -1.0) <= 1e+264) {
		tmp = fma(log(y), (x + -1.0), y) - t;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -1e+208)
		tmp = Float64(Float64(y * Float64(z * fma(y, -0.5, -1.0))) - t);
	elseif (Float64(z + -1.0) <= 1e+264)
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -1e+208], N[(N[(y * N[(z * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(z + -1.0), $MachinePrecision], 1e+264], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -9.9999999999999998e207

   1. Initial program 43.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 43.8

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 43.8

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 82.9%

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

   if -9.9999999999999998e207 < (-.f64 z #s(literal 1 binary64)) < 1.00000000000000004e264

   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

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.0%

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

   if 1.00000000000000004e264 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 24.6%

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

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -1 \cdot 10^{+208}:\\ \;\;\;\;y \cdot \left(z \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{elif}\;z + -1 \leq 10^{+264}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -1 \cdot 10^{+208}:\\ \;\;\;\;y \cdot \left(z \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{elif}\;z + -1 \leq 10^{+264}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -1e+208)
   (- (* y (* z (fma y -0.5 -1.0))) t)
   (if (<= (+ z -1.0) 1e+264) (- (* (log y) (+ x -1.0)) t) (- (* y (- z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -1e+208) {
		tmp = (y * (z * fma(y, -0.5, -1.0))) - t;
	} else if ((z + -1.0) <= 1e+264) {
		tmp = (log(y) * (x + -1.0)) - t;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -1e+208)
		tmp = Float64(Float64(y * Float64(z * fma(y, -0.5, -1.0))) - t);
	elseif (Float64(z + -1.0) <= 1e+264)
		tmp = Float64(Float64(log(y) * Float64(x + -1.0)) - t);
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -1e+208], N[(N[(y * N[(z * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(z + -1.0), $MachinePrecision], 1e+264], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -9.9999999999999998e207

   1. Initial program 43.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 43.8

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 43.8

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 82.9%

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

   if -9.9999999999999998e207 < (-.f64 z #s(literal 1 binary64)) < 1.00000000000000004e264

   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. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if 1.00000000000000004e264 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 24.6%

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

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -1 \cdot 10^{+208}:\\ \;\;\;\;y \cdot \left(z \cdot \mathsf{fma}\left(y, -0.5, -1\right)\right) - t\\ \mathbf{elif}\;z + -1 \leq 10^{+264}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -4.8e+57)
     t_1
     (if (<= x -7.2e-47)
       (- (* (log1p (- y)) z) t)
       (if (<= x 1.0) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -4.8e+57) {
		tmp = t_1;
	} else if (x <= -7.2e-47) {
		tmp = (log1p(-y) * z) - t;
	} else if (x <= 1.0) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -4.8e+57) {
		tmp = t_1;
	} else if (x <= -7.2e-47) {
		tmp = (Math.log1p(-y) * z) - t;
	} else if (x <= 1.0) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -4.8e+57:
		tmp = t_1
	elif x <= -7.2e-47:
		tmp = (math.log1p(-y) * z) - t
	elif x <= 1.0:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -4.8e+57)
		tmp = t_1;
	elseif (x <= -7.2e-47)
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	elseif (x <= 1.0)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -4.8e+57], t$95$1, If[LessEqual[x, -7.2e-47], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 1.0], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.80000000000000009e57 or 1 < x

   1. Initial program 91.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -4.80000000000000009e57 < x < -7.19999999999999982e-47

   1. Initial program 68.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -7.19999999999999982e-47 < x < 1

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 86.9

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 86.9

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

   4. Applied rewrites 86.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
   6. 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. mul-1-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 61.8

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

   7. Applied rewrites 61.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 85.8

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

   10. Applied rewrites 85.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 84.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+57}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -7 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-47}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+24}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -7e+80)
     t_1
     (if (<= x -7.2e-47)
       (-
        (* z (* y (fma y (fma y (fma y -0.25 -0.3333333333333333) -0.5) -1.0)))
        t)
       (if (<= x 6e+24) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -7e+80) {
		tmp = t_1;
	} else if (x <= -7.2e-47) {
		tmp = (z * (y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t;
	} else if (x <= 6e+24) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -7e+80)
		tmp = t_1;
	elseif (x <= -7.2e-47)
		tmp = Float64(Float64(z * Float64(y * fma(y, fma(y, fma(y, -0.25, -0.3333333333333333), -0.5), -1.0))) - t);
	elseif (x <= 6e+24)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+80], t$95$1, If[LessEqual[x, -7.2e-47], N[(N[(z * N[(y * N[(y * N[(y * N[(y * -0.25 + -0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 6e+24], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.99999999999999987e80 or 5.9999999999999999e24 < x

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -6.99999999999999987e80 < x < -7.19999999999999982e-47

   1. Initial program 75.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 75.2

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 75.2

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

   4. Applied rewrites 75.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
   6. 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. mul-1-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 82.9

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

   7. Applied rewrites 82.9%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 78.4%

       \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right) - t \]

   if -7.19999999999999982e-47 < x < 5.9999999999999999e24

   1. Initial program 86.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 86.1

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-+.f64 N/A

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

      15. metadata-eval 86.1

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

   4. Applied rewrites 86.1%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
   6. 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. mul-1-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 85.0

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

   10. Applied rewrites 85.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 82.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+80}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-47}:\\ \;\;\;\;z \cdot \left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)\right) - t\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+24}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

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

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

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 98.7

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Alternative 13: 66.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -4.5e+81) t_1 (if (<= x 8e+26) (- (fma y (- z) y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -4.5e+81) {
		tmp = t_1;
	} else if (x <= 8e+26) {
		tmp = fma(y, -z, y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -4.5e+81)
		tmp = t_1;
	elseif (x <= 8e+26)
		tmp = Float64(fma(y, Float64(-z), y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e+81], t$95$1, If[LessEqual[x, 8e+26], N[(N[(y * (-z) + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000017e81 or 8.00000000000000038e26 < x

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -4.50000000000000017e81 < x < 8.00000000000000038e26

   1. Initial program 84.3%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.0%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.7% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 87.7

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-+.f64 N/A

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

   15. metadata-eval 87.7

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

4. Applied rewrites 87.7%

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

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
6. 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. mul-1-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 49.9

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

7. Applied rewrites 49.9%

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

8. Taylor expanded in y around 0

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

10. Applied rewrites 49.3%

    \[\leadsto z \cdot \left(y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\right) - t \]
11. Add Preprocessing

Alternative 15: 46.5% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

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

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

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 49.2%

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

Alternative 16: 46.3% accurate, 20.5× speedup

Math

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

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(y * Float64(-z)) - t)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

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

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

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 49.0%

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

Alternative 17: 36.0% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in 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 37.6

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

5. Applied rewrites 37.6%

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

Reproduce

herbie shell --seed 2024233 
(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))