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

Percentage Accurate: 88.9% → 99.8%

Time: 15.6s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

   1. sub-neg 91.2%

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

   2. +-commutative 91.2%

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

   3. associate-+l+ 91.2%

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

   4. fma-def 91.2%

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

   5. sub-neg 91.2%

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

   6. metadata-eval 91.2%

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

   7. sub-neg 91.2%

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

   8. log1p-def 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. sub-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(1 - z\right)\\ \mathbf{if}\;-1 + x \leq -50 \lor \neg \left(-1 + x \leq -1\right):\\ \;\;\;\;\left(t_1 + x \cdot \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- 1.0 z))))
   (if (or (<= (+ -1.0 x) -50.0) (not (<= (+ -1.0 x) -1.0)))
     (- (+ t_1 (* x (log y))) t)
     (- (- t_1 (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (1.0 - z);
	double tmp;
	if (((-1.0 + x) <= -50.0) || !((-1.0 + x) <= -1.0)) {
		tmp = (t_1 + (x * log(y))) - t;
	} else {
		tmp = (t_1 - 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) :: t_1
    real(8) :: tmp
    t_1 = y * (1.0d0 - z)
    if ((((-1.0d0) + x) <= (-50.0d0)) .or. (.not. (((-1.0d0) + x) <= (-1.0d0)))) then
        tmp = (t_1 + (x * log(y))) - t
    else
        tmp = (t_1 - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (1.0 - z);
	double tmp;
	if (((-1.0 + x) <= -50.0) || !((-1.0 + x) <= -1.0)) {
		tmp = (t_1 + (x * Math.log(y))) - t;
	} else {
		tmp = (t_1 - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (1.0 - z)
	tmp = 0
	if ((-1.0 + x) <= -50.0) or not ((-1.0 + x) <= -1.0):
		tmp = (t_1 + (x * math.log(y))) - t
	else:
		tmp = (t_1 - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(1.0 - z))
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -50.0) || !(Float64(-1.0 + x) <= -1.0))
		tmp = Float64(Float64(t_1 + Float64(x * log(y))) - t);
	else
		tmp = Float64(Float64(t_1 - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (1.0 - z);
	tmp = 0.0;
	if (((-1.0 + x) <= -50.0) || ~(((-1.0 + x) <= -1.0)))
		tmp = (t_1 + (x * log(y))) - t;
	else
		tmp = (t_1 - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -50.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0]], $MachinePrecision]], N[(N[(t$95$1 + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(t$95$1 - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -50 or -1 < (-.f64 x 1)

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

      5. mul-1-neg 99.3%

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

      6. +-commutative 99.3%

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

      7. distribute-lft-in 99.3%

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

      8. metadata-eval 99.3%

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

      9. neg-mul-1 99.3%

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

      10. unsub-neg 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around inf 99.3%

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

      1. *-commutative 99.3%

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

   7. Simplified 99.3%

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

   if -50 < (-.f64 x 1) < -1

   1. Initial program 87.1%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. mul-1-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-lft-in 99.5%

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

      8. metadata-eval 99.5%

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

      9. neg-mul-1 99.5%

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

      10. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. mul-1-neg 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in y around 0 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. +-commutative 98.8%

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

      4. log-rec 98.8%

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

      5. unsub-neg 98.8%

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

   10. Simplified 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -50 \lor \neg \left(-1 + x \leq -1\right):\\ \;\;\;\;\left(y \cdot \left(1 - z\right) + x \cdot \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]

Alternative 3: 95.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+15} \lor \neg \left(-1 + x \leq 200000000000\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -5e+15) (not (<= (+ -1.0 x) 200000000000.0)))
   (- (* x (log 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+15) || !((-1.0 + x) <= 200000000000.0)) {
		tmp = (x * log(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+15)) .or. (.not. (((-1.0d0) + x) <= 200000000000.0d0))) then
        tmp = (x * log(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+15) || !((-1.0 + x) <= 200000000000.0)) {
		tmp = (x * Math.log(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+15) or not ((-1.0 + x) <= 200000000000.0):
		tmp = (x * math.log(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+15) || !(Float64(-1.0 + x) <= 200000000000.0))
		tmp = Float64(Float64(x * log(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+15) || ~(((-1.0 + x) <= 200000000000.0)))
		tmp = (x * log(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+15], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], 200000000000.0]], $MachinePrecision]], N[(N[(x * N[Log[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 1) < -5e15 or 2e11 < (-.f64 x 1)

   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. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

      5. mul-1-neg 99.3%

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

      6. +-commutative 99.3%

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

      7. distribute-lft-in 99.3%

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

      8. metadata-eval 99.3%

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

      9. neg-mul-1 99.3%

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

      10. unsub-neg 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around inf 94.8%

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

      1. *-commutative 94.8%

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

   7. Simplified 94.8%

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

   if -5e15 < (-.f64 x 1) < 2e11

   1. Initial program 86.8%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. mul-1-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-lft-in 99.5%

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

      8. metadata-eval 99.5%

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

      9. neg-mul-1 99.5%

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

      10. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. mul-1-neg 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. +-commutative 98.9%

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

      4. log-rec 98.9%

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

      5. unsub-neg 98.9%

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

   10. Simplified 98.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+15} \lor \neg \left(-1 + x \leq 200000000000\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]

Alternative 4: 95.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq 200000000000:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\frac{\log y}{\frac{1}{-1 + x}} - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -5e+15)
   (- (* x (log y)) t)
   (if (<= (+ -1.0 x) 200000000000.0)
     (- (- (* y (- 1.0 z)) (log y)) t)
     (- (/ (log y) (/ 1.0 (+ -1.0 x))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5e+15) {
		tmp = (x * log(y)) - t;
	} else if ((-1.0 + x) <= 200000000000.0) {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	} else {
		tmp = (log(y) / (1.0 / (-1.0 + x))) - 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+15)) then
        tmp = (x * log(y)) - t
    else if (((-1.0d0) + x) <= 200000000000.0d0) then
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    else
        tmp = (log(y) / (1.0d0 / ((-1.0d0) + x))) - 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+15) {
		tmp = (x * Math.log(y)) - t;
	} else if ((-1.0 + x) <= 200000000000.0) {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	} else {
		tmp = (Math.log(y) / (1.0 / (-1.0 + x))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -5e+15:
		tmp = (x * math.log(y)) - t
	elif (-1.0 + x) <= 200000000000.0:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	else:
		tmp = (math.log(y) / (1.0 / (-1.0 + x))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -5e+15)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(-1.0 + x) <= 200000000000.0)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	else
		tmp = Float64(Float64(log(y) / Float64(1.0 / Float64(-1.0 + x))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -5e+15)
		tmp = (x * log(y)) - t;
	elseif ((-1.0 + x) <= 200000000000.0)
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	else
		tmp = (log(y) / (1.0 / (-1.0 + x))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -5e+15], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 200000000000.0], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] / N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -5e15

   1. Initial program 97.8%

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

   2. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. mul-1-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. distribute-lft-in 99.8%

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

      8. metadata-eval 99.8%

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

      9. neg-mul-1 99.8%

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

      10. unsub-neg 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in x around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

   if -5e15 < (-.f64 x 1) < 2e11

   1. Initial program 86.8%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. mul-1-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-lft-in 99.5%

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

      8. metadata-eval 99.5%

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

      9. neg-mul-1 99.5%

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

      10. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. mul-1-neg 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in y around 0 98.9%

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

      1. mul-1-neg 98.9%

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

      2. log-rec 98.9%

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

      3. +-commutative 98.9%

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

      4. log-rec 98.9%

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

      5. unsub-neg 98.9%

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

   10. Simplified 98.9%

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

   if 2e11 < (-.f64 x 1)

   1. Initial program 93.6%

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

      1. sub-neg 93.6%

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

      2. +-commutative 93.6%

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

      3. associate-+l+ 93.6%

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

      4. fma-def 93.6%

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

      5. sub-neg 93.6%

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

      6. metadata-eval 93.6%

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

      7. sub-neg 93.6%

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

      8. log1p-def 99.6%

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

      9. sub-neg 99.6%

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

      10. sub-neg 99.6%

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 92.0%

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

      1. *-commutative 92.0%

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

      2. sub-neg 92.0%

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

      3. metadata-eval 92.0%

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

      4. /-rgt-identity 92.0%

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

      5. metadata-eval 92.0%

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

      6. sub-neg 92.0%

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

      7. *-commutative 92.0%

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

      8. associate-/l* 92.1%

         \[\leadsto \color{blue}{\frac{\log y}{\frac{1}{x - 1}}} - t \]

      9. sub-neg 92.1%

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

      10. metadata-eval 92.1%

          \[\leadsto \frac{\log y}{\frac{1}{x + \color{blue}{-1}}} - t \]

   6. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{\frac{\log y}{\frac{1}{x + -1}}} - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq 200000000000:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\frac{\log y}{\frac{1}{-1 + x}} - t\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

2. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. distribute-rgt-neg-in 99.4%

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

   5. mul-1-neg 99.4%

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

   6. +-commutative 99.4%

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

   7. distribute-lft-in 99.4%

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

   8. metadata-eval 99.4%

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

   9. neg-mul-1 99.4%

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

   10. unsub-neg 99.4%

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

4. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 6: 89.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+239}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+243}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.02e+239)
   (- (* z (- y)) t)
   (if (<= z 2.8e+243)
     (- (* (log y) (+ -1.0 x)) t)
     (- (* z (log1p (- y))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.02e+239) {
		tmp = (z * -y) - t;
	} else if (z <= 2.8e+243) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.02e+239) {
		tmp = (z * -y) - t;
	} else if (z <= 2.8e+243) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.02e+239:
		tmp = (z * -y) - t
	elif z <= 2.8e+243:
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.02e+239)
		tmp = Float64(Float64(z * Float64(-y)) - t);
	elseif (z <= 2.8e+243)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.02e+239], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 2.8e+243], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.02e239

   1. Initial program 35.2%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. mul-1-neg 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. neg-mul-1 100.0%

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

      10. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around inf 88.8%

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

      1. mul-1-neg 88.8%

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

      2. distribute-rgt-neg-in 88.8%

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

   7. Simplified 88.8%

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

   if -1.02e239 < z < 2.7999999999999999e243

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      2. +-commutative 95.0%

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

      3. associate-+l+ 95.0%

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

      4. fma-def 95.0%

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

      5. sub-neg 95.0%

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

      6. metadata-eval 95.0%

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

      7. sub-neg 95.0%

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

      8. log1p-def 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. sub-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 94.7%

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

   if 2.7999999999999999e243 < z

   1. Initial program 57.3%

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

   2. Taylor expanded in x around 0 42.8%

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

      1. +-commutative 42.8%

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

      2. mul-1-neg 42.8%

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

      3. unsub-neg 42.8%

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

      4. sub-neg 42.8%

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

      5. mul-1-neg 42.8%

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

      6. log1p-def 84.6%

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

      7. mul-1-neg 84.6%

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

      8. sub-neg 84.6%

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

      9. metadata-eval 84.6%

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

      10. +-commutative 84.6%

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

   4. Simplified 84.6%

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

   5. Taylor expanded in z around inf 43.1%

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

      1. sub-neg 43.1%

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

      2. log1p-def 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+239}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+243}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 7: 86.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -18500000000 \lor \neg \left(x \leq 7.6 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -18500000000.0) (not (<= x 7.6e-19)))
   (- (* x (log y)) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -18500000000.0) || !(x <= 7.6e-19)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-18500000000.0d0)) .or. (.not. (x <= 7.6d-19))) then
        tmp = (x * log(y)) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -18500000000.0) || !(x <= 7.6e-19)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -18500000000.0) or not (x <= 7.6e-19):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -18500000000.0) || !(x <= 7.6e-19))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -18500000000.0) || ~((x <= 7.6e-19)))
		tmp = (x * log(y)) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -18500000000.0], N[Not[LessEqual[x, 7.6e-19]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85e10 or 7.6e-19 < 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. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

      5. mul-1-neg 99.3%

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

      6. +-commutative 99.3%

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

      7. distribute-lft-in 99.3%

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

      8. metadata-eval 99.3%

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

      9. neg-mul-1 99.3%

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

      10. unsub-neg 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around inf 94.9%

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

      1. *-commutative 94.9%

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

   7. Simplified 94.9%

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

   if -1.85e10 < x < 7.6e-19

   1. Initial program 86.4%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. mul-1-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-lft-in 99.5%

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

      8. metadata-eval 99.5%

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

      9. neg-mul-1 99.5%

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

      10. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in x around 0 98.8%

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

      1. mul-1-neg 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in y around 0 85.1%

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

      1. mul-1-neg 85.1%

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

   10. Simplified 85.1%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -18500000000 \lor \neg \left(x \leq 7.6 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]

Alternative 8: 59.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+170} \lor \neg \left(z \leq 2.05 \cdot 10^{+36}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -4.5e+170) (not (<= z 2.05e+36)))
   (- (* z (- y)) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.5e+170) || !(z <= 2.05e+36)) {
		tmp = (z * -y) - t;
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-4.5d+170)) .or. (.not. (z <= 2.05d+36))) then
        tmp = (z * -y) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.5e+170) || !(z <= 2.05e+36)) {
		tmp = (z * -y) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -4.5e+170) or not (z <= 2.05e+36):
		tmp = (z * -y) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -4.5e+170) || !(z <= 2.05e+36))
		tmp = Float64(Float64(z * Float64(-y)) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -4.5e+170) || ~((z <= 2.05e+36)))
		tmp = (z * -y) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -4.5e+170], N[Not[LessEqual[z, 2.05e+36]], $MachinePrecision]], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.50000000000000022e170 or 2.05000000000000006e36 < z

   1. Initial program 74.8%

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

   2. Taylor expanded in y around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. sub-neg 98.4%

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

      3. metadata-eval 98.4%

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

      4. distribute-rgt-neg-in 98.4%

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

      5. mul-1-neg 98.4%

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

      6. +-commutative 98.4%

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

      7. distribute-lft-in 98.4%

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

      8. metadata-eval 98.4%

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

      9. neg-mul-1 98.4%

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

      10. unsub-neg 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in z around inf 58.5%

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

      1. mul-1-neg 58.5%

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

      2. distribute-rgt-neg-in 58.5%

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

   7. Simplified 58.5%

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

   if -4.50000000000000022e170 < z < 2.05000000000000006e36

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. mul-1-neg 99.9%

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

      6. +-commutative 99.9%

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

      7. distribute-lft-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. neg-mul-1 99.9%

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

      10. unsub-neg 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 64.3%

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

      1. mul-1-neg 64.3%

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

   7. Simplified 64.3%

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

   8. Taylor expanded in y around 0 63.8%

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

      1. mul-1-neg 63.8%

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

   10. Simplified 63.8%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+170} \lor \neg \left(z \leq 2.05 \cdot 10^{+36}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]

Alternative 9: 41.4% accurate, 26.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.3 \cdot 10^{-59} \lor \neg \left(t \leq 1.4 \cdot 10^{-22}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -7.3e-59) (not (<= t 1.4e-22))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.3e-59) || !(t <= 1.4e-22)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-7.3d-59)) .or. (.not. (t <= 1.4d-22))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.3e-59) || !(t <= 1.4e-22)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -7.3e-59) or not (t <= 1.4e-22):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -7.3e-59) || !(t <= 1.4e-22))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -7.3e-59) || ~((t <= 1.4e-22)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -7.3e-59], N[Not[LessEqual[t, 1.4e-22]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.3000000000000004e-59 or 1.39999999999999997e-22 < t

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-+l+ 99.4%

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

      4. fma-def 99.4%

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

      5. sub-neg 99.4%

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

      6. metadata-eval 99.4%

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

      7. sub-neg 99.4%

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

      8. log1p-def 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. sub-neg 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 75.1%

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

      1. neg-mul-1 75.1%

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

   6. Simplified 75.1%

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

   if -7.3000000000000004e-59 < t < 1.39999999999999997e-22

   1. Initial program 83.6%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. mul-1-neg 99.1%

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

      6. +-commutative 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. neg-mul-1 99.1%

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

      10. unsub-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in z around inf 18.5%

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

      1. mul-1-neg 18.5%

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

      2. distribute-rgt-neg-in 18.5%

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

   7. Simplified 18.5%

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

   8. Taylor expanded in y around inf 18.4%

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

      1. mul-1-neg 18.4%

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

      2. distribute-rgt-neg-in 18.4%

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

   10. Simplified 18.4%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.3 \cdot 10^{-59} \lor \neg \left(t \leq 1.4 \cdot 10^{-22}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 10: 41.3% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-59}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-22}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7e-59) (- (* z y) t) (if (<= t 1.02e-22) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7e-59) {
		tmp = (z * y) - t;
	} else if (t <= 1.02e-22) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-7d-59)) then
        tmp = (z * y) - t
    else if (t <= 1.02d-22) then
        tmp = z * -y
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7e-59) {
		tmp = (z * y) - t;
	} else if (t <= 1.02e-22) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7e-59:
		tmp = (z * y) - t
	elif t <= 1.02e-22:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7e-59)
		tmp = Float64(Float64(z * y) - t);
	elseif (t <= 1.02e-22)
		tmp = Float64(z * Float64(-y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7e-59)
		tmp = (z * y) - t;
	elseif (t <= 1.02e-22)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7e-59], N[(N[(z * y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 1.02e-22], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < -7.0000000000000002e-59

   1. Initial program 99.2%

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

   2. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

      5. mul-1-neg 99.4%

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

      6. +-commutative 99.4%

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

      7. distribute-lft-in 99.4%

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

      8. metadata-eval 99.4%

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

      9. neg-mul-1 99.4%

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

      10. unsub-neg 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in z around inf 73.3%

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

      1. mul-1-neg 73.3%

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

      2. distribute-rgt-neg-in 73.3%

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

   7. Simplified 73.3%

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

      1. sub-neg 73.3%

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

      2. *-commutative 73.3%

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

      3. add-sqr-sqrt 34.1%

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

      4. sqrt-unprod 63.1%

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

      5. sqr-neg 63.1%

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

      6. sqrt-unprod 38.3%

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

      7. add-sqr-sqrt 72.4%

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

   9. Applied egg-rr 72.4%

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

   if -7.0000000000000002e-59 < t < 1.02000000000000002e-22

   1. Initial program 83.6%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

      2. sub-neg 99.1%

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

      3. metadata-eval 99.1%

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

      4. distribute-rgt-neg-in 99.1%

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

      5. mul-1-neg 99.1%

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

      6. +-commutative 99.1%

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

      7. distribute-lft-in 99.1%

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

      8. metadata-eval 99.1%

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

      9. neg-mul-1 99.1%

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

      10. unsub-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in z around inf 18.5%

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

      1. mul-1-neg 18.5%

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

      2. distribute-rgt-neg-in 18.5%

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

   7. Simplified 18.5%

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

   8. Taylor expanded in y around inf 18.4%

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

      1. mul-1-neg 18.4%

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

      2. distribute-rgt-neg-in 18.4%

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

   10. Simplified 18.4%

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

   if 1.02000000000000002e-22 < t

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. fma-def 99.7%

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

      5. sub-neg 99.7%

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

      6. metadata-eval 99.7%

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

      7. sub-neg 99.7%

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

      8. log1p-def 100.0%

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

      9. sub-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 78.8%

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

      1. neg-mul-1 78.8%

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

   6. Simplified 78.8%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-59}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-22}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 45.7% 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 91.2%

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

2. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. distribute-rgt-neg-in 99.4%

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

   5. mul-1-neg 99.4%

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

   6. +-commutative 99.4%

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

   7. distribute-lft-in 99.4%

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

   8. metadata-eval 99.4%

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

   9. neg-mul-1 99.4%

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

   10. unsub-neg 99.4%

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

4. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. distribute-lft-in 99.4%

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

6. Applied egg-rr 99.4%

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

   1. distribute-lft-out 99.4%

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

   2. metadata-eval 99.4%

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

   3. sub-neg 99.4%

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

   4. flip-- 77.6%

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

   5. metadata-eval 77.6%

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

   6. fma-neg 77.6%

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

   7. metadata-eval 77.6%

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

   8. +-commutative 77.6%

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

   9. associate-*r/ 77.6%

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

   10. *-commutative 77.6%

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

   11. clear-num 77.5%

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

   12. clear-num 77.5%

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

   13. associate-*l/ 77.6%

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

   14. metadata-eval 77.6%

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

   15. fma-neg 77.6%

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

   16. metadata-eval 77.6%

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

   17. +-commutative 77.6%

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

   18. flip-- 99.3%

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

   19. sub-neg 99.3%

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

   20. metadata-eval 99.3%

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

8. Applied egg-rr 99.3%

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

9. Taylor expanded in y around inf 46.2%

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

10. Final simplification 46.2%

    \[\leadsto y \cdot \left(1 - z\right) - t \]

Alternative 12: 45.5% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

2. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. distribute-rgt-neg-in 99.4%

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

   5. mul-1-neg 99.4%

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

   6. +-commutative 99.4%

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

   7. distribute-lft-in 99.4%

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

   8. metadata-eval 99.4%

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

   9. neg-mul-1 99.4%

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

   10. unsub-neg 99.4%

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

4. Simplified 99.4%

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

5. Taylor expanded in z around inf 46.0%

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

   1. mul-1-neg 46.0%

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

   2. distribute-rgt-neg-in 46.0%

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

7. Simplified 46.0%

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

8. Final simplification 46.0%

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

Alternative 13: 34.9% 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 91.2%

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

   1. sub-neg 91.2%

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

   2. +-commutative 91.2%

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

   3. associate-+l+ 91.2%

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

   4. fma-def 91.2%

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

   5. sub-neg 91.2%

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

   6. metadata-eval 91.2%

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

   7. sub-neg 91.2%

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

   8. log1p-def 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. sub-neg 99.9%

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

   10. sub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around inf 37.5%

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

   1. neg-mul-1 37.5%

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

6. Simplified 37.5%

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

7. Final simplification 37.5%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023318 
(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))