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

Percentage Accurate: 88.9% → 99.3%

Time: 14.5s

Alternatives: 17

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(-y\right)\\ t_2 := \mathsf{fma}\left(\log y, x - 1, \left(1 - z\right) \cdot t\_1\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(t\_1, z - 1, \left(x - 1\right) \cdot \log y\right), {t\_2}^{-1} \cdot t\_2, -t\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (log1p (- y))) (t_2 (fma (log y) (- x 1.0) (* (- 1.0 z) t_1))))
   (fma
    (fma t_1 (- z 1.0) (* (- x 1.0) (log y)))
    (* (pow t_2 -1.0) t_2)
    (- t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log1p(-y);
	double t_2 = fma(log(y), (x - 1.0), ((1.0 - z) * t_1));
	return fma(fma(t_1, (z - 1.0), ((x - 1.0) * log(y))), (pow(t_2, -1.0) * t_2), -t);
}

Julia

function code(x, y, z, t)
	t_1 = log1p(Float64(-y))
	t_2 = fma(log(y), Float64(x - 1.0), Float64(Float64(1.0 - z) * t_1))
	return fma(fma(t_1, Float64(z - 1.0), Float64(Float64(x - 1.0) * log(y))), Float64((t_2 ^ -1.0) * t_2), Float64(-t))
end

Wolfram

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

Derivation

1. Initial program 88.4%

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y\\ \mathbf{if}\;t\_2 \leq -100000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 255:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t))
        (t_2 (+ (* (log (- 1.0 y)) (- z 1.0)) (* (- x 1.0) (log y)))))
   (if (<= t_2 -100000000.0)
     t_1
     (if (<= t_2 255.0)
       (- (* (- 1.0 z) y) t)
       (if (<= t_2 1000.0) (- (- y (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * log(y));
	double tmp;
	if (t_2 <= -100000000.0) {
		tmp = t_1;
	} else if (t_2 <= 255.0) {
		tmp = ((1.0 - z) * y) - t;
	} else if (t_2 <= 1000.0) {
		tmp = (y - 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 = (x * log(y)) - t
    t_2 = (log((1.0d0 - y)) * (z - 1.0d0)) + ((x - 1.0d0) * log(y))
    if (t_2 <= (-100000000.0d0)) then
        tmp = t_1
    else if (t_2 <= 255.0d0) then
        tmp = ((1.0d0 - z) * y) - t
    else if (t_2 <= 1000.0d0) then
        tmp = (y - 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 = (x * Math.log(y)) - t;
	double t_2 = (Math.log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * Math.log(y));
	double tmp;
	if (t_2 <= -100000000.0) {
		tmp = t_1;
	} else if (t_2 <= 255.0) {
		tmp = ((1.0 - z) * y) - t;
	} else if (t_2 <= 1000.0) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	t_2 = (math.log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * math.log(y))
	tmp = 0
	if t_2 <= -100000000.0:
		tmp = t_1
	elif t_2 <= 255.0:
		tmp = ((1.0 - z) * y) - t
	elif t_2 <= 1000.0:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(Float64(x - 1.0) * log(y)))
	tmp = 0.0
	if (t_2 <= -100000000.0)
		tmp = t_1;
	elseif (t_2 <= 255.0)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	elseif (t_2 <= 1000.0)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - t;
	t_2 = (log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * log(y));
	tmp = 0.0;
	if (t_2 <= -100000000.0)
		tmp = t_1;
	elseif (t_2 <= 255.0)
		tmp = ((1.0 - z) * y) - t;
	elseif (t_2 <= 1000.0)
		tmp = (y - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -100000000.0], t$95$1, If[LessEqual[t$95$2, 255.0], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 1000.0], N[(N[(y - N[Log[y], $MachinePrecision]), $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)))) < -1e8 or 1e3 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 93.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if -1e8 < (+.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)))) < 255

   1. Initial program 67.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

   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 75.7%

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

   if 255 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 1e3

   1. Initial program 93.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 92.8%

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

   12. Taylor expanded in y around 0

       \[\leadsto \left(y - \log y\right) - t \]
   13. Step-by-step derivation

   14. Applied rewrites 92.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq -100000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 255:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 1000:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 255:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+54}:\\ \;\;\;\;-\left(t + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (+ (* (log (- 1.0 y)) (- z 1.0)) (* (- x 1.0) (log y)))))
   (if (<= t_2 -2e+88)
     t_1
     (if (<= t_2 255.0)
       (- (* (- 1.0 z) y) t)
       (if (<= t_2 5e+54) (- (+ t (log y))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = (log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * log(y));
	double tmp;
	if (t_2 <= -2e+88) {
		tmp = t_1;
	} else if (t_2 <= 255.0) {
		tmp = ((1.0 - z) * y) - t;
	} else if (t_2 <= 5e+54) {
		tmp = -(t + log(y));
	} 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 = x * log(y)
    t_2 = (log((1.0d0 - y)) * (z - 1.0d0)) + ((x - 1.0d0) * log(y))
    if (t_2 <= (-2d+88)) then
        tmp = t_1
    else if (t_2 <= 255.0d0) then
        tmp = ((1.0d0 - z) * y) - t
    else if (t_2 <= 5d+54) then
        tmp = -(t + log(y))
    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 = x * Math.log(y);
	double t_2 = (Math.log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * Math.log(y));
	double tmp;
	if (t_2 <= -2e+88) {
		tmp = t_1;
	} else if (t_2 <= 255.0) {
		tmp = ((1.0 - z) * y) - t;
	} else if (t_2 <= 5e+54) {
		tmp = -(t + Math.log(y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = (math.log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * math.log(y))
	tmp = 0
	if t_2 <= -2e+88:
		tmp = t_1
	elif t_2 <= 255.0:
		tmp = ((1.0 - z) * y) - t
	elif t_2 <= 5e+54:
		tmp = -(t + math.log(y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(Float64(x - 1.0) * log(y)))
	tmp = 0.0
	if (t_2 <= -2e+88)
		tmp = t_1;
	elseif (t_2 <= 255.0)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	elseif (t_2 <= 5e+54)
		tmp = Float64(-Float64(t + log(y)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = (log((1.0 - y)) * (z - 1.0)) + ((x - 1.0) * log(y));
	tmp = 0.0;
	if (t_2 <= -2e+88)
		tmp = t_1;
	elseif (t_2 <= 255.0)
		tmp = ((1.0 - z) * y) - t;
	elseif (t_2 <= 5e+54)
		tmp = -(t + log(y));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+88], t$95$1, If[LessEqual[t$95$2, 255.0], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 5e+54], (-N[(t + N[Log[y], $MachinePrecision]), $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)))) < -1.99999999999999992e88 or 5.00000000000000005e54 < (+.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 94.7%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 71.9

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

   7. Applied rewrites 71.9%

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

   if -1.99999999999999992e88 < (+.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)))) < 255

   1. Initial program 72.9%

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

   3. Taylor expanded in y around 0

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

      1. *-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 98.2

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

   5. Applied rewrites 98.2%

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

   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 72.9%

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

   if 255 < (+.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)))) < 5.00000000000000005e54

   1. Initial program 91.5%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq -2 \cdot 10^{+88}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 255:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \left(x - 1\right) \cdot \log y \leq 5 \cdot 10^{+54}:\\ \;\;\;\;-\left(t + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

4. Applied rewrites 99.6%

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

Alternative 5: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift--.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. neg-mul-1 N/A

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

   9. lift-neg.f64 N/A

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

   10. lower-fma.f64 99.4

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

7. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate--l+ N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

9. Applied rewrites 99.4%

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

Alternative 6: 99.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-neg.f64 N/A

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

   6. associate-+l+ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. lower-fma.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 7: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

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

Alternative 8: 95.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{elif}\;x - 1 \leq -0.5:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x 1.0) -2e+24)
   (fma (- x 1.0) (log y) (- t))
   (if (<= (- x 1.0) -0.5)
     (- (- (fma (- z 1.0) y (log y))) t)
     (- (fma (log y) (- x 1.0) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - 1.0) <= -2e+24) {
		tmp = fma((x - 1.0), log(y), -t);
	} else if ((x - 1.0) <= -0.5) {
		tmp = -fma((z - 1.0), y, log(y)) - t;
	} else {
		tmp = fma(log(y), (x - 1.0), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -2e+24)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	elseif (Float64(x - 1.0) <= -0.5)
		tmp = Float64(Float64(-fma(Float64(z - 1.0), y, log(y))) - t);
	else
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - 1.0), $MachinePrecision], -2e+24], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.5], N[((-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision]) - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2e24

   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

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 95.8

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

   5. Applied rewrites 95.8%

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

   if -2e24 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 82.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.7%

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

   if -0.5 < (-.f64 x #s(literal 1 binary64))

   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 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 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.6%

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

Alternative 9: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 10: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 1e+222)
   (- (fma (log y) (- x 1.0) y) t)
   (-
    (* (* (fma (fma (fma -0.25 y -0.3333333333333333) y -0.5) y -1.0) y) z)
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 1e+222) {
		tmp = fma(log(y), (x - 1.0), y) - t;
	} else {
		tmp = ((fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 1e+222)
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+222], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 1e222

   1. Initial program 91.9%

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

   3. Taylor expanded in y around 0

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

      1. *-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 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.7%

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

   if 1e222 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 52.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 1e+222)
   (fma (- x 1.0) (log y) (- t))
   (-
    (* (* (fma (fma (fma -0.25 y -0.3333333333333333) y -0.5) y -1.0) y) z)
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 1e+222) {
		tmp = fma((x - 1.0), log(y), -t);
	} else {
		tmp = ((fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 1e+222)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	else
		tmp = Float64(Float64(Float64(fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+222], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 1e222

   1. Initial program 91.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if 1e222 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 52.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 72.0%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 99.2% 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 88.4%

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

Alternative 13: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 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 z around inf

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

8. Applied rewrites 98.8%

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

Alternative 14: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+83}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.3e+52)
     t_1
     (if (<= x 9.5e+83)
       (-
        (* (* (fma (fma (fma -0.25 y -0.3333333333333333) y -0.5) y -1.0) y) z)
        t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.3e+52) {
		tmp = t_1;
	} else if (x <= 9.5e+83) {
		tmp = ((fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.3e+52)
		tmp = t_1;
	elseif (x <= 9.5e+83)
		tmp = Float64(Float64(Float64(fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e+52], t$95$1, If[LessEqual[x, 9.5e+83], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e52 or 9.5000000000000002e83 < x

   1. Initial program 96.3%

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   6. 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 75.4

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

   7. Applied rewrites 75.4%

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

   if -1.3e52 < x < 9.5000000000000002e83

   1. Initial program 83.4%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 63.9

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

   5. Applied rewrites 63.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+83}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.1% 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 88.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 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 48.5%

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

Alternative 16: 45.9% 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 88.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 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 z around inf

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

8. Applied rewrites 48.2%

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

9. Final simplification 48.2%

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

Alternative 17: 35.4% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 37.0

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

5. Applied rewrites 37.0%

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

Reproduce

herbie shell --seed 2024331 
(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))