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

Percentage Accurate: 89.5% → 99.8%

Time: 14.4s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 93.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - 1, \log y, y - t\right)\\ t_2 := \left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t\\ \mathbf{if}\;t\_2 \leq 40:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 624.5:\\ \;\;\;\;\mathsf{fma}\left(-1, \log y, z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- x 1.0) (log y) (- y t)))
        (t_2 (- (+ (* (log (- 1.0 y)) (- z 1.0)) (* (log y) (- x 1.0))) t)))
   (if (<= t_2 40.0)
     t_1
     (if (<= t_2 624.5) (fma -1.0 (log y) (* z (- y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((x - 1.0), log(y), (y - t));
	double t_2 = ((log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0))) - t;
	double tmp;
	if (t_2 <= 40.0) {
		tmp = t_1;
	} else if (t_2 <= 624.5) {
		tmp = fma(-1.0, log(y), (z * -y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(x - 1.0), log(y), Float64(y - t))
	t_2 = Float64(Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(log(y) * Float64(x - 1.0))) - t)
	tmp = 0.0
	if (t_2 <= 40.0)
		tmp = t_1;
	elseif (t_2 <= 624.5)
		tmp = fma(-1.0, log(y), Float64(z * Float64(-y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$2, 40.0], t$95$1, If[LessEqual[t$95$2, 624.5], N[(-1.0 * N[Log[y], $MachinePrecision] + N[(z * (-y)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < 40 or 624.5 < (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t)

   1. Initial program 96.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 95.9%

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

   if 40 < (-.f64 (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) t) < 624.5

   1. Initial program 68.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t \leq 40:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \mathbf{elif}\;\left(\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\right) - t \leq 624.5:\\ \;\;\;\;\mathsf{fma}\left(-1, \log y, z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -1.000002:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{elif}\;x - 1 \leq -0.99999999999995:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x 1.0) -1.000002)
   (fma (- x 1.0) (log y) (- t))
   (if (<= (- x 1.0) -0.99999999999995)
     (- (fma (- 1.0 z) y (- (log y))) t)
     (fma (- x 1.0) (log y) (- y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - 1.0) <= -1.000002) {
		tmp = fma((x - 1.0), log(y), -t);
	} else if ((x - 1.0) <= -0.99999999999995) {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	} else {
		tmp = fma((x - 1.0), log(y), (y - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1.000002)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	elseif (Float64(x - 1.0) <= -0.99999999999995)
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(y - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.000002], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.99999999999995], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.00000200000000006

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -1.00000200000000006 < (-.f64 x #s(literal 1 binary64)) < -0.99999999999995004

   1. Initial program 85.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in y around 0

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.9%

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

   if -0.99999999999995004 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 94.0%

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

Alternative 4: 96.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -430000000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot x\right) - t\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -430000000.0)
   (- (fma (- 1.0 z) y (* (log y) x)) t)
   (if (<= t 2e-27)
     (fma (- x 1.0) (log y) (* z (- y)))
     (fma (- x 1.0) (log y) (- y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -430000000.0) {
		tmp = fma((1.0 - z), y, (log(y) * x)) - t;
	} else if (t <= 2e-27) {
		tmp = fma((x - 1.0), log(y), (z * -y));
	} else {
		tmp = fma((x - 1.0), log(y), (y - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -430000000.0)
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(log(y) * x)) - t);
	elseif (t <= 2e-27)
		tmp = fma(Float64(x - 1.0), log(y), Float64(z * Float64(-y)));
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(y - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -430000000.0], N[(N[(N[(1.0 - z), $MachinePrecision] * y + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 2e-27], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(z * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.3e8

   1. Initial program 96.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 96.4

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

   4. Applied rewrites 96.4%

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

   5. Taylor expanded in y around 0

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.9%

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

   if -4.3e8 < t < 2.0000000000000001e-27

   1. Initial program 83.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 99.2%

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

   if 2.0000000000000001e-27 < t

   1. Initial program 97.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 97.4%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -430000000:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot x\right) - t\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.2e-12)
   (fma (- x 1.0) (log y) (- t))
   (if (<= t 2e-27)
     (fma (- x 1.0) (log y) (* z (- y)))
     (fma (- x 1.0) (log y) (- y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.2e-12) {
		tmp = fma((x - 1.0), log(y), -t);
	} else if (t <= 2e-27) {
		tmp = fma((x - 1.0), log(y), (z * -y));
	} else {
		tmp = fma((x - 1.0), log(y), (y - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.2e-12)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	elseif (t <= 2e-27)
		tmp = fma(Float64(x - 1.0), log(y), Float64(z * Float64(-y)));
	else
		tmp = fma(Float64(x - 1.0), log(y), Float64(y - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.2e-12], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[t, 2e-27], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(z * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.19999999999999994e-12

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -1.19999999999999994e-12 < t < 2.0000000000000001e-27

   1. Initial program 82.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 99.3%

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

   if 2.0000000000000001e-27 < t

   1. Initial program 97.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 97.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= (- x 1.0) -2e+64)
     t_1
     (if (<= (- x 1.0) 1e+16) (- (fma (- y) z y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((x - 1.0) <= -2e+64) {
		tmp = t_1;
	} else if ((x - 1.0) <= 1e+16) {
		tmp = fma(-y, z, y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (Float64(x - 1.0) <= -2e+64)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= 1e+16)
		tmp = Float64(fma(Float64(-y), z, y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -2e+64], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], 1e+16], N[(N[((-y) * z + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2.00000000000000004e64 or 1e16 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -2.00000000000000004e64 < (-.f64 x #s(literal 1 binary64)) < 1e16

   1. Initial program 86.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in y around 0

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 63.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x - 1 \leq 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 2e+206) (fma (- x 1.0) (log y) (- y t)) (- (* (- y) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 2e+206) {
		tmp = fma((x - 1.0), log(y), (y - t));
	} else {
		tmp = (-y * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 2e+206)
		tmp = fma(Float64(x - 1.0), log(y), Float64(y - t));
	else
		tmp = Float64(Float64(Float64(-y) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 2e+206], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 2.0000000000000001e206

   1. Initial program 94.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 94.0%

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

   if 2.0000000000000001e206 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 43.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 89.7

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -1.66 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 11.5:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= x -1.66e-6) t_1 (if (<= x 11.5) (- (fma (- y) z y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if (x <= -1.66e-6) {
		tmp = t_1;
	} else if (x <= 11.5) {
		tmp = fma(-y, z, y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	tmp = 0.0
	if (x <= -1.66e-6)
		tmp = t_1;
	elseif (x <= 11.5)
		tmp = Float64(fma(Float64(-y), z, y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.66e-6], t$95$1, If[LessEqual[x, 11.5], N[(N[((-y) * z + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65999999999999999e-6 or 11.5 < 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

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -1.65999999999999999e-6 < x < 11.5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 85.4

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in y around 0

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 65.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.66 \cdot 10^{-6}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq 11.5:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 2e+206) (fma (- x 1.0) (log y) (- t)) (- (* (- y) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 2e+206) {
		tmp = fma((x - 1.0), log(y), -t);
	} else {
		tmp = (-y * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 2e+206)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	else
		tmp = Float64(Float64(Float64(-y) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 2e+206], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 2.0000000000000001e206

   1. Initial program 94.1%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 93.8

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

   5. Applied rewrites 93.8%

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

   if 2.0000000000000001e206 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 43.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 89.7

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.7%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 11: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. associate--r+ N/A

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 12: 43.4% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -115000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 62000:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -115000.0) (- t) (if (<= t 62000.0) (* (- y) z) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -115000.0) {
		tmp = -t;
	} else if (t <= 62000.0) {
		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 <= (-115000.0d0)) then
        tmp = -t
    else if (t <= 62000.0d0) 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 <= -115000.0) {
		tmp = -t;
	} else if (t <= 62000.0) {
		tmp = -y * z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -115000.0:
		tmp = -t
	elif t <= 62000.0:
		tmp = -y * z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -115000.0)
		tmp = Float64(-t);
	elseif (t <= 62000.0)
		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 <= -115000.0)
		tmp = -t;
	elseif (t <= 62000.0)
		tmp = -y * z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -115000.0], (-t), If[LessEqual[t, 62000.0], N[((-y) * z), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -115000 or 62000 < t

   1. Initial program 96.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if -115000 < t < 62000

   1. Initial program 84.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

      10. lift--.f64 N/A

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

      11. lower-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 17.2

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

   7. Applied rewrites 17.2%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 17.1%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -115000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 62000:\\ \;\;\;\;\left(-y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. flip3-- N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. clear-num N/A

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

   9. flip3-- N/A

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

   10. lift--.f64 N/A

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

   11. lower-/.f64 90.3

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

4. Applied rewrites 90.3%

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

5. Taylor expanded in y around 0

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. sub-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 44.4%

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

Alternative 14: 46.2% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 44.2%

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

9. Final simplification 44.2%

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

Alternative 15: 36.2% 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 90.4%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.6

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

5. Applied rewrites 35.6%

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

Reproduce

herbie shell --seed 2024249 
(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))