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

Percentage Accurate: 89.4% → 99.8%

Time: 27.3s

Alternatives: 22

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. sub-neg 91.4%

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

   2. +-commutative 91.4%

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

   3. associate-+l+ 91.4%

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

   4. fma-define 91.4%

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

   5. sub-neg 91.4%

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

   6. metadata-eval 91.4%

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

   7. sub-neg 91.4%

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

   8. log1p-define 99.9%

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

   9. fma-define 99.9%

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

   10. sub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. +-commutative 91.4%

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

   2. fma-define 91.4%

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

   3. sub-neg 91.4%

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

   4. metadata-eval 91.4%

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

   5. sub-neg 91.4%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)))))) - 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 = ((log(y) * ((-1.0d0) + x)) + ((z + (-1.0d0)) * (y * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)))))) - t
end function

Java

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

Python

def code(x, y, z, t):
	return ((math.log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)))))) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)))))) - t;
end

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \left(\log y \cdot \left(-1 + x\right) + \left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right)\right) - t \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - \log y\right) - t\\ t_2 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -10000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-214}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-287}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- y (log y)) t)) (t_2 (- (* x (log y)) t)))
   (if (<= x -10000000000.0)
     t_2
     (if (<= x -1.16e-214)
       t_1
       (if (<= x -6.2e-287)
         (- (* y (* z (+ -1.0 (* y -0.5)))) t)
         (if (<= x 6.3e-53) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - log(y)) - t;
	double t_2 = (x * log(y)) - t;
	double tmp;
	if (x <= -10000000000.0) {
		tmp = t_2;
	} else if (x <= -1.16e-214) {
		tmp = t_1;
	} else if (x <= -6.2e-287) {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	} else if (x <= 6.3e-53) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y - log(y)) - t
    t_2 = (x * log(y)) - t
    if (x <= (-10000000000.0d0)) then
        tmp = t_2
    else if (x <= (-1.16d-214)) then
        tmp = t_1
    else if (x <= (-6.2d-287)) then
        tmp = (y * (z * ((-1.0d0) + (y * (-0.5d0))))) - t
    else if (x <= 6.3d-53) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - Math.log(y)) - t;
	double t_2 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -10000000000.0) {
		tmp = t_2;
	} else if (x <= -1.16e-214) {
		tmp = t_1;
	} else if (x <= -6.2e-287) {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	} else if (x <= 6.3e-53) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - math.log(y)) - t
	t_2 = (x * math.log(y)) - t
	tmp = 0
	if x <= -10000000000.0:
		tmp = t_2
	elif x <= -1.16e-214:
		tmp = t_1
	elif x <= -6.2e-287:
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t
	elif x <= 6.3e-53:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - log(y)) - t)
	t_2 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -10000000000.0)
		tmp = t_2;
	elseif (x <= -1.16e-214)
		tmp = t_1;
	elseif (x <= -6.2e-287)
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * -0.5)))) - t);
	elseif (x <= 6.3e-53)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - log(y)) - t;
	t_2 = (x * log(y)) - t;
	tmp = 0.0;
	if (x <= -10000000000.0)
		tmp = t_2;
	elseif (x <= -1.16e-214)
		tmp = t_1;
	elseif (x <= -6.2e-287)
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	elseif (x <= 6.3e-53)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -10000000000.0], t$95$2, If[LessEqual[x, -1.16e-214], t$95$1, If[LessEqual[x, -6.2e-287], N[(N[(y * N[(z * N[(-1.0 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 6.3e-53], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e10 or 6.29999999999999979e-53 < x

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0 99.8%

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

   4. Taylor expanded in x around inf 93.2%

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

      1. *-commutative 93.2%

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

   6. Simplified 93.2%

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

   if -1e10 < x < -1.16e-214 or -6.2000000000000001e-287 < x < 6.29999999999999979e-53

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0 99.6%

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

   4. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in y around 0 97.6%

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

      1. associate-*r* 97.6%

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

      2. neg-mul-1 97.6%

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

      3. sub-neg 97.6%

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

      4. metadata-eval 97.6%

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

      5. +-commutative 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in z around 0 87.6%

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

   if -1.16e-214 < x < -6.2000000000000001e-287

   1. Initial program 70.9%

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

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in z around inf 79.4%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(-0.5 \cdot y - 1\right)\right)} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-214}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-287}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-53}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-\log y\right) - t\\ t_2 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -10000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-286}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (log y)) t)) (t_2 (- (* x (log y)) t)))
   (if (<= x -10000000000.0)
     t_2
     (if (<= x -6.4e-215)
       t_1
       (if (<= x -3.9e-286)
         (- (* y (* z (+ -1.0 (* y -0.5)))) t)
         (if (<= x 5.4e-30) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -log(y) - t;
	double t_2 = (x * log(y)) - t;
	double tmp;
	if (x <= -10000000000.0) {
		tmp = t_2;
	} else if (x <= -6.4e-215) {
		tmp = t_1;
	} else if (x <= -3.9e-286) {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	} else if (x <= 5.4e-30) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -log(y) - t
    t_2 = (x * log(y)) - t
    if (x <= (-10000000000.0d0)) then
        tmp = t_2
    else if (x <= (-6.4d-215)) then
        tmp = t_1
    else if (x <= (-3.9d-286)) then
        tmp = (y * (z * ((-1.0d0) + (y * (-0.5d0))))) - t
    else if (x <= 5.4d-30) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -Math.log(y) - t;
	double t_2 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -10000000000.0) {
		tmp = t_2;
	} else if (x <= -6.4e-215) {
		tmp = t_1;
	} else if (x <= -3.9e-286) {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	} else if (x <= 5.4e-30) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -math.log(y) - t
	t_2 = (x * math.log(y)) - t
	tmp = 0
	if x <= -10000000000.0:
		tmp = t_2
	elif x <= -6.4e-215:
		tmp = t_1
	elif x <= -3.9e-286:
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t
	elif x <= 5.4e-30:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-log(y)) - t)
	t_2 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -10000000000.0)
		tmp = t_2;
	elseif (x <= -6.4e-215)
		tmp = t_1;
	elseif (x <= -3.9e-286)
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * -0.5)))) - t);
	elseif (x <= 5.4e-30)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -log(y) - t;
	t_2 = (x * log(y)) - t;
	tmp = 0.0;
	if (x <= -10000000000.0)
		tmp = t_2;
	elseif (x <= -6.4e-215)
		tmp = t_1;
	elseif (x <= -3.9e-286)
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	elseif (x <= 5.4e-30)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -10000000000.0], t$95$2, If[LessEqual[x, -6.4e-215], t$95$1, If[LessEqual[x, -3.9e-286], N[(N[(y * N[(z * N[(-1.0 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 5.4e-30], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e10 or 5.39999999999999975e-30 < x

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0 99.8%

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

   4. Taylor expanded in x around inf 94.7%

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

      1. *-commutative 94.7%

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

   6. Simplified 94.7%

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

   if -1e10 < x < -6.4000000000000003e-215 or -3.89999999999999995e-286 < x < 5.39999999999999975e-30

   1. Initial program 89.7%

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

   3. Taylor expanded in y around 0 99.6%

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

   4. Taylor expanded in x around 0 98.8%

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

      1. mul-1-neg 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in y around 0 86.7%

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

      1. neg-mul-1 86.7%

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

   9. Simplified 86.7%

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

   if -6.4000000000000003e-215 < x < -3.89999999999999995e-286

   1. Initial program 70.9%

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

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in z around inf 79.4%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(-0.5 \cdot y - 1\right)\right)} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-215}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{-286}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-30}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -5e+14) (not (<= (+ -1.0 x) -1.0)))
   (- (- (* x (log y)) (* z y)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -5e+14) || !((-1.0 + x) <= -1.0)) {
		tmp = ((x * log(y)) - (z * y)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-5d+14)) .or. (.not. (((-1.0d0) + x) <= (-1.0d0)))) then
        tmp = ((x * log(y)) - (z * y)) - t
    else
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -5e+14) || !((-1.0 + x) <= -1.0)) {
		tmp = ((x * Math.log(y)) - (z * y)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -5e+14) or not ((-1.0 + x) <= -1.0):
		tmp = ((x * math.log(y)) - (z * y)) - t
	else:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -5e+14) || !(Float64(-1.0 + x) <= -1.0))
		tmp = Float64(Float64(Float64(x * log(y)) - Float64(z * y)) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -5e+14) || ~(((-1.0 + x) <= -1.0)))
		tmp = ((x * log(y)) - (z * y)) - t;
	else
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -5e+14], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5e14 or -1 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. fma-define 99.7%

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

      5. mul-1-neg 99.7%

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

      6. fma-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. sub-neg 99.7%

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

      9. metadata-eval 99.7%

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

      10. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf 99.7%

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

   7. Taylor expanded in x around inf 99.3%

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

      1. *-commutative 99.3%

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

   9. Simplified 99.3%

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

   if -5e14 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 86.7%

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

   3. Taylor expanded in y around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. fma-define 98.6%

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

      5. mul-1-neg 98.6%

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

      6. fma-neg 98.6%

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

      7. +-commutative 98.6%

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

      8. sub-neg 98.6%

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

      9. metadata-eval 98.6%

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

      10. +-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in x around 0 97.2%

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

      1. neg-mul-1 97.2%

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

      2. cancel-sign-sub-inv 97.2%

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

      3. sub-neg 97.2%

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

      4. metadata-eval 97.2%

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

      5. +-commutative 97.2%

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

      6. *-commutative 97.2%

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

      7. +-commutative 97.2%

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

      8. sub-neg 97.2%

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

      9. *-commutative 97.2%

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

      10. distribute-lft-neg-in 97.2%

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

      11. distribute-rgt-neg-in 97.2%

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

      12. distribute-neg-in 97.2%

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

      13. metadata-eval 97.2%

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

      14. sub-neg 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 98.2%

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

Alternative 7: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -10000000000 \lor \neg \left(-1 + x \leq -1\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -10000000000.0) (not (<= (+ -1.0 x) -1.0)))
   (- (* (log y) (+ -1.0 x)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -10000000000.0) || !((-1.0 + x) <= -1.0)) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-10000000000.0d0)) .or. (.not. (((-1.0d0) + x) <= (-1.0d0)))) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -10000000000.0) || !((-1.0 + x) <= -1.0)) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -10000000000.0) or not ((-1.0 + x) <= -1.0):
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -10000000000.0) || !(Float64(-1.0 + x) <= -1.0))
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -10000000000.0) || ~(((-1.0 + x) <= -1.0)))
		tmp = (log(y) * (-1.0 + x)) - t;
	else
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -10000000000.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e10 or -1 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.1%

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

      1. +-commutative 97.1%

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

      2. fma-define 97.1%

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

      3. sub-neg 97.1%

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

      4. metadata-eval 97.1%

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

      5. sub-neg 97.1%

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

      6. log1p-define 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 96.9%

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

   if -1e10 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 86.6%

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

   3. Taylor expanded in y around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. fma-define 98.6%

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

      5. mul-1-neg 98.6%

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

      6. fma-neg 98.6%

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

      7. +-commutative 98.6%

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

      8. sub-neg 98.6%

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

      9. metadata-eval 98.6%

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

      10. +-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in x around 0 97.9%

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

      1. neg-mul-1 97.9%

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

      2. cancel-sign-sub-inv 97.9%

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

      3. sub-neg 97.9%

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

      4. metadata-eval 97.9%

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

      5. +-commutative 97.9%

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

      6. *-commutative 97.9%

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

      7. +-commutative 97.9%

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

      8. sub-neg 97.9%

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

      9. *-commutative 97.9%

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

      10. distribute-lft-neg-in 97.9%

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

      11. distribute-rgt-neg-in 97.9%

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

      12. distribute-neg-in 97.9%

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

      13. metadata-eval 97.9%

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

      14. sub-neg 97.9%

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

   8. Simplified 97.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -10000000000 \lor \neg \left(-1 + x \leq -1\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))))) - 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 = ((log(y) * ((-1.0d0) + x)) + ((z + (-1.0d0)) * (y * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0)))))) - t
end function

Java

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

Python

def code(x, y, z, t):
	return ((math.log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))))) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))))) - t;
end

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \left(\log y \cdot \left(-1 + x\right) + \left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right)\right) - t \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((log(y) * ((-1.0d0) + x)) + (y * (((-0.5d0) * (y * (z + (-1.0d0)))) + (1.0d0 - z)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(y) * (-1.0 + x)) + (y * ((-0.5 * (y * (z + -1.0))) + (1.0 - z)))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (-1.0 + x)) + (y * ((-0.5 * (y * (z + -1.0))) + (1.0 - z)))) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(y) * (-1.0 + x)) + (y * ((-0.5 * (y * (z + -1.0))) + (1.0 - z)))) - t;
end

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \left(\log y \cdot \left(-1 + x\right) + y \cdot \left(-0.5 \cdot \left(y \cdot \left(z + -1\right)\right) + \left(1 - z\right)\right)\right) - t \]
5. Add Preprocessing

Alternative 10: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * -0.5))))) - 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 = ((log(y) * ((-1.0d0) + x)) + ((z + (-1.0d0)) * (y * ((-1.0d0) + (y * (-0.5d0)))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * -0.5))))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * -0.5))))) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * -0.5))))) - t;
end

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \left(\log y \cdot \left(-1 + x\right) + \left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot -0.5\right)\right)\right) - t \]
5. Add Preprocessing

Alternative 11: 60.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+194}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+92}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2e+194)
   (- (* y (- (* -0.5 (* z y)) z)) t)
   (if (<= z 2.2e+92)
     (- (- (log y)) t)
     (- (* y (* z (+ -1.0 (* y (- (* y -0.3333333333333333) 0.5))))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e+194) {
		tmp = (y * ((-0.5 * (z * y)) - z)) - t;
	} else if (z <= 2.2e+92) {
		tmp = -log(y) - t;
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2d+194)) then
        tmp = (y * (((-0.5d0) * (z * y)) - z)) - t
    else if (z <= 2.2d+92) then
        tmp = -log(y) - t
    else
        tmp = (y * (z * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e+194) {
		tmp = (y * ((-0.5 * (z * y)) - z)) - t;
	} else if (z <= 2.2e+92) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2e+194:
		tmp = (y * ((-0.5 * (z * y)) - z)) - t
	elif z <= 2.2e+92:
		tmp = -math.log(y) - t
	else:
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2e+194)
		tmp = Float64(Float64(y * Float64(Float64(-0.5 * Float64(z * y)) - z)) - t);
	elseif (z <= 2.2e+92)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2e+194)
		tmp = (y * ((-0.5 * (z * y)) - z)) - t;
	elseif (z <= 2.2e+92)
		tmp = -log(y) - t;
	else
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2e+194], N[(N[(y * N[(N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 2.2e+92], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.99999999999999989e194

   1. Initial program 65.7%

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

   3. Taylor expanded in y around 0 99.7%

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

   4. Taylor expanded in z around inf 56.7%

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

   5. Taylor expanded in y around inf 23.1%

      \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(-1 \cdot \frac{z}{y} + -0.5 \cdot z\right)\right)} - t \]
   6. Step-by-step derivation

      1. +-commutative 23.1%

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(-0.5 \cdot z + -1 \cdot \frac{z}{y}\right)}\right) - t \]

      2. mul-1-neg 23.1%

         \[\leadsto y \cdot \left(y \cdot \left(-0.5 \cdot z + \color{blue}{\left(-\frac{z}{y}\right)}\right)\right) - t \]

      3. unsub-neg 23.1%

         \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(-0.5 \cdot z - \frac{z}{y}\right)}\right) - t \]

   7. Simplified 23.1%

      \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(-0.5 \cdot z - \frac{z}{y}\right)\right)} - t \]

   8. Taylor expanded in y around 0 56.8%

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot z + -0.5 \cdot \left(y \cdot z\right)\right)} - t \]
   9. Step-by-step derivation

      1. +-commutative 56.8%

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

      2. mul-1-neg 56.8%

         \[\leadsto y \cdot \left(-0.5 \cdot \left(y \cdot z\right) + \color{blue}{\left(-z\right)}\right) - t \]

      3. sub-neg 56.8%

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

      4. *-commutative 56.8%

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

   10. Simplified 56.8%

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

   if -1.99999999999999989e194 < z < 2.19999999999999992e92

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0 99.7%

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

   4. Taylor expanded in x around 0 63.3%

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

      1. mul-1-neg 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in y around 0 61.9%

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

      1. neg-mul-1 61.9%

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

   9. Simplified 61.9%

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

   if 2.19999999999999992e92 < z

   1. Initial program 77.5%

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

   3. Taylor expanded in y around 0 99.9%

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

   4. Taylor expanded in y around 0 99.9%

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

   5. Taylor expanded in z around inf 56.4%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+194}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+92}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. sub-neg 99.1%

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

   3. metadata-eval 99.1%

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

   4. fma-define 99.1%

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

   5. mul-1-neg 99.1%

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

   6. fma-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. sub-neg 99.1%

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

   9. metadata-eval 99.1%

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

   10. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 13: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (-1.0 + x)) - (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 = ((log(y) * ((-1.0d0) + x)) - (z * y)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. sub-neg 99.1%

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

   3. metadata-eval 99.1%

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

   4. fma-define 99.1%

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

   5. mul-1-neg 99.1%

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

   6. fma-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. sub-neg 99.1%

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

   9. metadata-eval 99.1%

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

   10. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in z around inf 99.0%

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

7. Final simplification 99.0%

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

Alternative 14: 88.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (log(y) * (-1.0 + x)) - 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 = (log(y) * ((-1.0d0) + x)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

   1. +-commutative 91.4%

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

   2. fma-define 91.4%

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

   3. sub-neg 91.4%

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

   4. metadata-eval 91.4%

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

   5. sub-neg 91.4%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around 0 90.2%

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

6. Final simplification 90.2%

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

Alternative 15: 46.6% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * (z * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0))))) - t
end function

Java

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

Python

def code(x, y, z, t):
	return (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
end

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.7%

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

4. Taylor expanded in y around 0 99.6%

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

5. Taylor expanded in z around inf 47.8%

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

6. Final simplification 47.8%

   \[\leadsto y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t \]
7. Add Preprocessing

Alternative 16: 42.8% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-73} \lor \neg \left(t \leq 8\right):\\ \;\;\;\;3 - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5.5e-73) (not (<= t 8.0))) (- 3.0 t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.5e-73) || !(t <= 8.0)) {
		tmp = 3.0 - t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-5.5d-73)) .or. (.not. (t <= 8.0d0))) then
        tmp = 3.0d0 - t
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.5e-73) || !(t <= 8.0)) {
		tmp = 3.0 - t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -5.5e-73) or not (t <= 8.0):
		tmp = 3.0 - t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5.5e-73) || !(t <= 8.0))
		tmp = Float64(3.0 - t);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -5.5e-73) || ~((t <= 8.0)))
		tmp = 3.0 - t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5.5e-73], N[Not[LessEqual[t, 8.0]], $MachinePrecision]], N[(3.0 - t), $MachinePrecision], N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.50000000000000006e-73 or 8 < t

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0 97.0%

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

      1. +-commutative 97.0%

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

      2. sub-neg 97.0%

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

      3. metadata-eval 97.0%

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

      4. fma-define 97.0%

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

      5. mul-1-neg 97.0%

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

      6. fma-neg 97.0%

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

      7. +-commutative 97.0%

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

      8. sub-neg 97.0%

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

      9. log1p-define 97.0%

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

   5. Simplified 97.0%

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

   7. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{3} - t \]

   if -5.50000000000000006e-73 < t < 8

   1. Initial program 82.6%

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

   3. Taylor expanded in y around 0 99.4%

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

   4. Taylor expanded in z around inf 20.4%

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

   5. Taylor expanded in y around 0 20.0%

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

      1. associate-*r* 20.0%

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

      2. neg-mul-1 20.0%

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

   7. Simplified 20.0%

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

   8. Taylor expanded in y around inf 19.9%

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

      1. neg-mul-1 19.9%

         \[\leadsto \color{blue}{-y \cdot z} \]

      2. distribute-rgt-neg-in 19.9%

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

   10. Simplified 19.9%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-73} \lor \neg \left(t \leq 8\right):\\ \;\;\;\;3 - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.5% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * (((-0.5d0) * (z * y)) - z)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * ((-0.5 * (z * y)) - z)) - t;
}

Python

def code(x, y, z, t):
	return (y * ((-0.5 * (z * y)) - z)) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * ((-0.5 * (z * y)) - z)) - t;
end

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.4%

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

4. Taylor expanded in z around inf 47.7%

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

5. Taylor expanded in y around inf 34.7%

   \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(-1 \cdot \frac{z}{y} + -0.5 \cdot z\right)\right)} - t \]
6. Step-by-step derivation

   1. +-commutative 34.7%

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(-0.5 \cdot z + -1 \cdot \frac{z}{y}\right)}\right) - t \]

   2. mul-1-neg 34.7%

      \[\leadsto y \cdot \left(y \cdot \left(-0.5 \cdot z + \color{blue}{\left(-\frac{z}{y}\right)}\right)\right) - t \]

   3. unsub-neg 34.7%

      \[\leadsto y \cdot \left(y \cdot \color{blue}{\left(-0.5 \cdot z - \frac{z}{y}\right)}\right) - t \]

7. Simplified 34.7%

   \[\leadsto y \cdot \color{blue}{\left(y \cdot \left(-0.5 \cdot z - \frac{z}{y}\right)\right)} - t \]

8. Taylor expanded in y around 0 47.7%

   \[\leadsto y \cdot \color{blue}{\left(-1 \cdot z + -0.5 \cdot \left(y \cdot z\right)\right)} - t \]
9. Step-by-step derivation

   1. +-commutative 47.7%

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

   2. mul-1-neg 47.7%

      \[\leadsto y \cdot \left(-0.5 \cdot \left(y \cdot z\right) + \color{blue}{\left(-z\right)}\right) - t \]

   3. sub-neg 47.7%

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

   4. *-commutative 47.7%

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

10. Simplified 47.7%

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

11. Final simplification 47.7%

    \[\leadsto y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t \]
12. Add Preprocessing

Alternative 18: 46.5% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * (z * ((-1.0d0) + (y * (-0.5d0))))) - t
end function

Java

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

Python

def code(x, y, z, t):
	return (y * (z * (-1.0 + (y * -0.5)))) - t

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
end

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.4%

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

4. Taylor expanded in z around inf 47.7%

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

5. Final simplification 47.7%

   \[\leadsto y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t \]
6. Add Preprocessing

Alternative 19: 46.5% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. sub-neg 99.1%

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

   3. metadata-eval 99.1%

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

   4. fma-define 99.1%

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

   5. mul-1-neg 99.1%

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

   6. fma-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. sub-neg 99.1%

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

   9. metadata-eval 99.1%

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

   10. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Taylor expanded in y around inf 47.6%

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

Alternative 20: 46.3% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in y around 0 99.4%

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

4. Taylor expanded in z around inf 47.7%

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

5. Taylor expanded in y around 0 47.4%

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

   1. associate-*r* 47.4%

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

   2. neg-mul-1 47.4%

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

7. Simplified 47.4%

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

8. Final simplification 47.4%

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

Alternative 21: 38.3% accurate, 71.7× speedup

Math

\[\begin{array}{l} \\ 3 - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 3.0 t))

C

double code(double x, double y, double z, double t) {
	return 3.0 - 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 = 3.0d0 - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return 3.0 - t;
}

Python

def code(x, y, z, t):
	return 3.0 - t

Julia

function code(x, y, z, t)
	return Float64(3.0 - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 3.0 - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(3.0 - t), $MachinePrecision]

Derivation

1. Initial program 91.4%

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

3. Taylor expanded in z around 0 90.7%

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

   1. +-commutative 90.7%

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

   2. sub-neg 90.7%

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

   3. metadata-eval 90.7%

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

   4. fma-define 90.7%

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

   5. mul-1-neg 90.7%

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

   6. fma-neg 90.7%

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

   7. +-commutative 90.7%

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

   8. sub-neg 90.7%

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

   9. log1p-define 90.7%

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

5. Simplified 90.7%

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

7. Applied egg-rr 41.1%

   \[\leadsto \color{blue}{3} - t \]
8. Add Preprocessing

Alternative 22: 36.1% accurate, 107.5× 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 91.4%

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

   1. +-commutative 91.4%

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

   2. fma-define 91.4%

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

   3. sub-neg 91.4%

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

   4. metadata-eval 91.4%

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

   5. sub-neg 91.4%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in t around inf 39.1%

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

   1. mul-1-neg 39.1%

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

7. Simplified 39.1%

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

Reproduce

herbie shell --seed 2024105 
(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))