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

Percentage Accurate: 89.2% → 99.8%

Time: 19.9s

Alternatives: 14

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

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

   6. sub-neg N/A

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

   7. accelerator-lowering-log1p.f64 N/A

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

   8. neg-lowering-neg.f64 N/A

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

   9. clear-num N/A

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

   10. flip-- N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sub-neg N/A

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

   13. +-lowering-+.f64 N/A

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

   14. metadata-eval 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f64 99.6

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 95.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x + -1, -t\right)\\ \mathbf{if}\;x + -1 \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq -0.999998:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, \left(-t\right) - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) (+ x -1.0) (- t))))
   (if (<= (+ x -1.0) -500000.0)
     t_1
     (if (<= (+ x -1.0) -0.999998) (fma y (- 1.0 z) (- (- t) (log y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), (x + -1.0), -t);
	double tmp;
	if ((x + -1.0) <= -500000.0) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.999998) {
		tmp = fma(y, (1.0 - z), (-t - log(y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), Float64(x + -1.0), Float64(-t))
	tmp = 0.0
	if (Float64(x + -1.0) <= -500000.0)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= -0.999998)
		tmp = fma(y, Float64(1.0 - z), Float64(Float64(-t) - log(y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -500000.0], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], -0.999998], N[(y * N[(1.0 - z), $MachinePrecision] + N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5e5 or -0.999998000000000054 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. neg-lowering-neg.f64 94.0

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

   5. Simplified 94.0%

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

   if -5e5 < (-.f64 x #s(literal 1 binary64)) < -0.999998000000000054

   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

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 93.9%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. --lowering--.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. --lowering--.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. log-lowering-log.f64 N/A

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. +-lowering-+.f64 99.6

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

   11. Simplified 99.6%

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

   12. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 N/A

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

       3. log-lowering-log.f64 99.6

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

   14. Simplified 99.6%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -500000:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \mathbf{elif}\;x + -1 \leq -0.999998:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, \left(-t\right) - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% 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 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   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. accelerator-lowering-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. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 89.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, x \cdot \log y\right)\\ \mathbf{elif}\;z + -1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -2e+252)
   (fma y (- 1.0 z) (* x (log y)))
   (if (<= (+ z -1.0) 5e+167)
     (- (fma (log y) (+ x -1.0) y) t)
     (- (* (log1p (- y)) z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -2e+252) {
		tmp = fma(y, (1.0 - z), (x * log(y)));
	} else if ((z + -1.0) <= 5e+167) {
		tmp = fma(log(y), (x + -1.0), y) - t;
	} else {
		tmp = (log1p(-y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -2e+252)
		tmp = fma(y, Float64(1.0 - z), Float64(x * log(y)));
	elseif (Float64(z + -1.0) <= 5e+167)
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	else
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -2e+252], N[(y * N[(1.0 - z), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z + -1.0), $MachinePrecision], 5e+167], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 z #s(literal 1 binary64)) < -2.0000000000000002e252

   1. Initial program 50.4%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 48.2%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. --lowering--.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. --lowering--.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. log-lowering-log.f64 N/A

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. +-lowering-+.f64 99.3

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

   11. Simplified 99.3%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 N/A

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

       3. log-lowering-log.f64 81.7

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

   14. Simplified 81.7%

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

   if -2.0000000000000002e252 < (-.f64 z #s(literal 1 binary64)) < 4.9999999999999997e167

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 87.9%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. --lowering--.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. --lowering--.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. log-lowering-log.f64 N/A

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. +-lowering-+.f64 99.7

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

   11. Simplified 99.7%

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

   12. Taylor expanded in z around 0

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

       1. --lowering--.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. log-lowering-log.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-lowering-+.f64 97.9

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

   14. Simplified 97.9%

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

   if 4.9999999999999997e167 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 52.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 77.9

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

   5. Simplified 77.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+252}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, x \cdot \log y\right)\\ \mathbf{elif}\;z + -1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) 5e+167)
   (- (fma (log y) (+ x -1.0) y) t)
   (- (* (log1p (- y)) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= 5e+167) {
		tmp = fma(log(y), (x + -1.0), y) - t;
	} else {
		tmp = (log1p(-y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= 5e+167)
		tmp = Float64(fma(log(y), Float64(x + -1.0), y) - t);
	else
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], 5e+167], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 4.9999999999999997e167

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 84.8%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. --lowering--.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. --lowering--.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. log-lowering-log.f64 N/A

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. +-lowering-+.f64 99.7

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

   11. Simplified 99.7%

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

   12. Taylor expanded in z around 0

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

       1. --lowering--.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. log-lowering-log.f64 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. +-lowering-+.f64 94.0

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

   14. Simplified 94.0%

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

   if 4.9999999999999997e167 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 52.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 77.9

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

   5. Simplified 77.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -0.000106:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 185:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -0.000106) t_1 (if (<= x 185.0) (fma y (- 1.0 z) (- 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 <= -0.000106) {
		tmp = t_1;
	} else if (x <= 185.0) {
		tmp = fma(y, (1.0 - z), -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 <= -0.000106)
		tmp = t_1;
	elseif (x <= 185.0)
		tmp = fma(y, Float64(1.0 - z), Float64(-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, -0.000106], t$95$1, If[LessEqual[x, 185.0], N[(y * N[(1.0 - z), $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.06e-4 or 185 < x

   1. Initial program 94.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 93.2

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

   5. Simplified 93.2%

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

   if -1.06e-4 < x < 185

   1. Initial program 85.4%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 94.0%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. --lowering--.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. --lowering--.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. log-lowering-log.f64 N/A

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. +-lowering-+.f64 99.6

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

   11. Simplified 99.6%

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

   12. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 62.9

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

   14. Simplified 62.9%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000106:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 185:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) 5e+167)
   (fma (log y) (+ x -1.0) (- t))
   (- (* (log1p (- y)) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= 5e+167) {
		tmp = fma(log(y), (x + -1.0), -t);
	} else {
		tmp = (log1p(-y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= 5e+167)
		tmp = fma(log(y), Float64(x + -1.0), Float64(-t));
	else
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], 5e+167], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 4.9999999999999997e167

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 N/A

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

      8. neg-lowering-neg.f64 93.8

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

   5. Simplified 93.8%

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

   if 4.9999999999999997e167 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 52.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 77.9

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

   5. Simplified 77.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x + -1, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% 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 90.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
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. accelerator-lowering-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. --lowering--.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. log-lowering-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. +-lowering-+.f64 99.4

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 10: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -4.4e+166) t_1 (if (<= x 1.85e+20) (fma y (- 1.0 z) (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -4.4e+166) {
		tmp = t_1;
	} else if (x <= 1.85e+20) {
		tmp = fma(y, (1.0 - z), -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 <= -4.4e+166)
		tmp = t_1;
	elseif (x <= 1.85e+20)
		tmp = fma(y, Float64(1.0 - z), Float64(-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, -4.4e+166], t$95$1, If[LessEqual[x, 1.85e+20], N[(y * N[(1.0 - z), $MachinePrecision] + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999998e166 or 1.85e20 < x

   1. Initial program 95.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 77.5

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

   5. Simplified 77.5%

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

   if -4.3999999999999998e166 < x < 1.85e20

   1. Initial program 87.1%

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

   3. Taylor expanded in y around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 99.5

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

   5. Simplified 99.5%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 91.8%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. accelerator-lowering-fma.f64 N/A

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

       3. --lowering--.f64 N/A

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

       4. +-commutative N/A

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

       5. distribute-neg-in N/A

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

       6. mul-1-neg N/A

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

       7. remove-double-neg N/A

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

       8. sub-neg N/A

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

       9. --lowering--.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. log-lowering-log.f64 N/A

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

       12. sub-neg N/A

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

       13. metadata-eval N/A

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

       14. +-lowering-+.f64 99.2

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

   11. Simplified 99.2%

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

   12. Taylor expanded in t around inf

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

       1. mul-1-neg N/A

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

       2. neg-lowering-neg.f64 61.1

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

   14. Simplified 61.1%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+166}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.4% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+18}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 270:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.6e+18) (- t) (if (<= t 270.0) (- (* y z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.6e+18) {
		tmp = -t;
	} else if (t <= 270.0) {
		tmp = -(y * z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-7.6d+18)) then
        tmp = -t
    else if (t <= 270.0d0) then
        tmp = -(y * z)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.6e+18) {
		tmp = -t;
	} else if (t <= 270.0) {
		tmp = -(y * z);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7.6e+18:
		tmp = -t
	elif t <= 270.0:
		tmp = -(y * z)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.6e+18)
		tmp = Float64(-t);
	elseif (t <= 270.0)
		tmp = Float64(-Float64(y * z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7.6e+18)
		tmp = -t;
	elseif (t <= 270.0)
		tmp = -(y * z);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.6e+18], (-t), If[LessEqual[t, 270.0], (-N[(y * z), $MachinePrecision]), (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -7.6e18 or 270 < t

   1. Initial program 94.9%

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

   3. Taylor expanded in 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. neg-lowering-neg.f64 73.8

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

   5. Simplified 73.8%

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

   if -7.6e18 < t < 270

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sub-neg N/A

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

      4. accelerator-lowering-log1p.f64 N/A

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

      5. neg-lowering-neg.f64 17.6

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

   5. Simplified 17.6%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 16.9

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

   8. Simplified 16.9%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.6 \cdot 10^{+18}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 270:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.7% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f64 99.6

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

5. Simplified 99.6%

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

6. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

8. Simplified 83.5%

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

9. Taylor expanded in y around 0

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

    1. sub-neg N/A

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

    2. accelerator-lowering-fma.f64 N/A

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

    3. --lowering--.f64 N/A

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

    4. +-commutative N/A

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

    5. distribute-neg-in N/A

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

    6. mul-1-neg N/A

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

    7. remove-double-neg N/A

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

    8. sub-neg N/A

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

    9. --lowering--.f64 N/A

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

    10. *-lowering-*.f64 N/A

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

    11. log-lowering-log.f64 N/A

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

    12. sub-neg N/A

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

    13. metadata-eval N/A

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

    14. +-lowering-+.f64 99.4

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

11. Simplified 99.4%

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

12. Taylor expanded in t around inf

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

    1. mul-1-neg N/A

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

    2. neg-lowering-neg.f64 46.8

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

14. Simplified 46.8%

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

Alternative 13: 46.5% accurate, 25.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in 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. *-lowering-*.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. accelerator-lowering-log1p.f64 N/A

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

   5. neg-lowering-neg.f64 47.0

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

5. Simplified 47.0%

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

6. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. neg-lowering-neg.f64 N/A

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

   5. accelerator-lowering-fma.f64 46.6

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

8. Simplified 46.6%

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

Alternative 14: 36.1% 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 90.1%

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

3. Taylor expanded in 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. neg-lowering-neg.f64 36.9

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

5. Simplified 36.9%

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

Reproduce

herbie shell --seed 2024199 
(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))