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

Percentage Accurate: 89.1% → 99.6%

Time: 14.0s

Alternatives: 13

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: 89.1% 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: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 108.5:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (<= t_2 -1e+130)
     t_1
     (if (<= t_2 108.5)
       (- (* (- 1.0 z) y) t)
       (if (<= t_2 2e+35) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if (t_2 <= -1e+130) {
		tmp = t_1;
	} else if (t_2 <= 108.5) {
		tmp = ((1.0 - z) * y) - t;
	} else if (t_2 <= 2e+35) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = ((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))
    if (t_2 <= (-1d+130)) then
        tmp = t_1
    else if (t_2 <= 108.5d0) then
        tmp = ((1.0d0 - z) * y) - t
    else if (t_2 <= 2d+35) then
        tmp = -log(y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double t_2 = ((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)));
	double tmp;
	if (t_2 <= -1e+130) {
		tmp = t_1;
	} else if (t_2 <= 108.5) {
		tmp = ((1.0 - z) * y) - t;
	} else if (t_2 <= 2e+35) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = ((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))
	tmp = 0
	if t_2 <= -1e+130:
		tmp = t_1
	elif t_2 <= 108.5:
		tmp = ((1.0 - z) * y) - t
	elif t_2 <= 2e+35:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_2 <= -1e+130)
		tmp = t_1;
	elseif (t_2 <= 108.5)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	elseif (t_2 <= 2e+35)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	tmp = 0.0;
	if (t_2 <= -1e+130)
		tmp = t_1;
	elseif (t_2 <= 108.5)
		tmp = ((1.0 - z) * y) - t;
	elseif (t_2 <= 2e+35)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -1e+130], t$95$1, If[LessEqual[t$95$2, 108.5], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 2e+35], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -1.0000000000000001e130 or 1.9999999999999999e35 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 93.8%

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

   3. Taylor expanded in y around 0

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

      1. *-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 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 74.7

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

   8. Applied rewrites 74.7%

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

   if -1.0000000000000001e130 < (+.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)))) < 108.5

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. *-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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 66.1%

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

   if 108.5 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 1.9999999999999999e35

   1. Initial program 89.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.0%

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

Alternative 3: 94.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+38}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x - 1 \leq -1:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x 1.0) -5e+38)
   (- (* (log y) x) t)
   (if (<= (- x 1.0) -1.0)
     (- (- (fma (- z 1.0) y (log y))) t)
     (- (* (- x 1.0) (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - 1.0) <= -5e+38) {
		tmp = (log(y) * x) - t;
	} else if ((x - 1.0) <= -1.0) {
		tmp = -fma((z - 1.0), y, log(y)) - t;
	} else {
		tmp = ((x - 1.0) * log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -5e+38)
		tmp = Float64(Float64(log(y) * x) - t);
	elseif (Float64(x - 1.0) <= -1.0)
		tmp = Float64(Float64(-fma(Float64(z - 1.0), y, log(y))) - t);
	else
		tmp = Float64(Float64(Float64(x - 1.0) * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+38], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.0], N[((-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision]) - t), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4.9999999999999997e38

   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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -4.9999999999999997e38 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 86.7%

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

   3. Taylor expanded in y around 0

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

      1. *-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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.7%

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

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

   1. Initial program 92.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 4: 95.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+20} \lor \neg \left(t \leq 16500000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, \left(1 - z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5.8e+20) (not (<= t 16500000.0)))
   (- (* (log y) x) t)
   (fma (log y) (- x 1.0) (* (- 1.0 z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.8e+20) || !(t <= 16500000.0)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = fma(log(y), (x - 1.0), ((1.0 - z) * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5.8e+20) || !(t <= 16500000.0))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = fma(log(y), Float64(x - 1.0), Float64(Float64(1.0 - z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5.8e+20], N[Not[LessEqual[t, 16500000.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.8e20 or 1.65e7 < t

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if -5.8e20 < t < 1.65e7

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-out N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in y around -inf

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

   11. Applied rewrites 98.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+20} \lor \neg \left(t \leq 16500000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, \left(1 - z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+20} \lor \neg \left(t \leq 16500000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, \mathsf{fma}\left(\log y, x - 1, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5.8e+20) (not (<= t 16500000.0)))
   (- (* (log y) x) t)
   (fma (- y) z (fma (log y) (- x 1.0) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.8e+20) || !(t <= 16500000.0)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = fma(-y, z, fma(log(y), (x - 1.0), y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5.8e+20) || !(t <= 16500000.0))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = fma(Float64(-y), z, fma(log(y), Float64(x - 1.0), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5.8e+20], N[Not[LessEqual[t, 16500000.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[((-y) * z + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.8e20 or 1.65e7 < t

   1. Initial program 96.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if -5.8e20 < t < 1.65e7

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-out N/A

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

      9. lower-neg.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 98.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+20} \lor \neg \left(t \leq 16500000\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, \mathsf{fma}\left(\log y, x - 1, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -41 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -41.0) (not (<= x 1.0))) (- (* (log y) x) t) (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -41.0) || !(x <= 1.0)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = -log(y) - 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 ((x <= (-41.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (log(y) * x) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -41.0) || !(x <= 1.0)) {
		tmp = (Math.log(y) * x) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -41.0) or not (x <= 1.0):
		tmp = (math.log(y) * x) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -41.0) || !(x <= 1.0))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -41.0) || ~((x <= 1.0)))
		tmp = (log(y) * x) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -41.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -41 or 1 < x

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -41 < x < 1

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 86.0

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -41 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-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 99.1

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

5. Applied rewrites 99.1%

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

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

6. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower--.f64 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-out N/A

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

   9. lower-neg.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 99.1

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

8. Applied rewrites 99.1%

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

Alternative 9: 66.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+34} \lor \neg \left(x \leq 3.1 \cdot 10^{+123}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.3e+34) (not (<= x 3.1e+123)))
   (* (log y) x)
   (- (* (- 1.0 z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.3e+34) || !(x <= 3.1e+123)) {
		tmp = log(y) * x;
	} else {
		tmp = ((1.0 - z) * y) - 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 ((x <= (-1.3d+34)) .or. (.not. (x <= 3.1d+123))) then
        tmp = log(y) * x
    else
        tmp = ((1.0d0 - z) * y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.3e+34) || !(x <= 3.1e+123)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = ((1.0 - z) * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.3e+34) or not (x <= 3.1e+123):
		tmp = math.log(y) * x
	else:
		tmp = ((1.0 - z) * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.3e+34) || !(x <= 3.1e+123))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.3e+34) || ~((x <= 3.1e+123)))
		tmp = log(y) * x;
	else
		tmp = ((1.0 - z) * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.3e+34], N[Not[LessEqual[x, 3.1e+123]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.29999999999999999e34 or 3.10000000000000006e123 < x

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0

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

      1. *-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 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 78.0

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

   8. Applied rewrites 78.0%

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

   if -1.29999999999999999e34 < x < 3.10000000000000006e123

   1. Initial program 87.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.9%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+34} \lor \neg \left(x \leq 3.1 \cdot 10^{+123}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2e-17) (- (* (- x 1.0) (log y)) t) (- (* (log1p (- y)) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2e-17) {
		tmp = ((x - 1.0) * log(y)) - t;
	} else {
		tmp = (log1p(-y) * z) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2e-17) {
		tmp = ((x - 1.0) * Math.log(y)) - t;
	} else {
		tmp = (Math.log1p(-y) * z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2e-17:
		tmp = ((x - 1.0) * math.log(y)) - t
	else:
		tmp = (math.log1p(-y) * z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2e-17)
		tmp = Float64(Float64(Float64(x - 1.0) * log(y)) - t);
	else
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2e-17], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000014e-17

   1. Initial program 91.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if 2.00000000000000014e-17 < y

   1. Initial program 75.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 86.3

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

   5. Applied rewrites 86.3%

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

Alternative 11: 46.2% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-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 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 47.4%

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

Alternative 12: 46.0% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in y around 0

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

   1. *-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 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 47.2%

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

Alternative 13: 35.6% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 36.8

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

5. Applied rewrites 36.8%

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

Reproduce

herbie shell --seed 2024318 
(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))