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

Percentage Accurate: 89.1% → 99.8%

Time: 16.2s

Alternatives: 16

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{log1p}\left(0 - y\right) \cdot \left(z + -1\right) + \log y \cdot \left(x + -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (log1p (- 0.0 y)) (+ z -1.0)) (* (log y) (+ x -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return ((log1p((0.0 - y)) * (z + -1.0)) + (log(y) * (x + -1.0))) - t;
}

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log1p((0.0 - y)) * (z + -1.0)) + (Math.log(y) * (x + -1.0))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log1p((0.0 - y)) * (z + -1.0)) + (math.log(y) * (x + -1.0))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log1p(Float64(0.0 - y)) * Float64(z + -1.0)) + Float64(log(y) * Float64(x + -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[1 + N[(0.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 92.1%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \log \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   3. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   5. --lowering--.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

5. Final simplification 99.8%

   \[\leadsto \left(\mathsf{log1p}\left(0 - y\right) \cdot \left(z + -1\right) + \log y \cdot \left(x + -1\right)\right) - t \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right)\right), t\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)\right), t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right)\right), t\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(-1 + y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{2} + \frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right)\right), t\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(y \cdot \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

   12. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 95.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log y}{\frac{1}{x + -1}} - t\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+19}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.15e-14)
   (- (/ (log y) (/ 1.0 (+ x -1.0))) t)
   (if (<= x 6.4e+19) (- (- (* y (- 1.0 z)) (log y)) t) (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.15e-14) {
		tmp = (log(y) / (1.0 / (x + -1.0))) - t;
	} else if (x <= 6.4e+19) {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	} else {
		tmp = (x * 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 <= (-1.15d-14)) then
        tmp = (log(y) / (1.0d0 / (x + (-1.0d0)))) - t
    else if (x <= 6.4d+19) then
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    else
        tmp = (x * 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 <= -1.15e-14) {
		tmp = (Math.log(y) / (1.0 / (x + -1.0))) - t;
	} else if (x <= 6.4e+19) {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	} else {
		tmp = (x * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.15e-14:
		tmp = (math.log(y) / (1.0 / (x + -1.0))) - t
	elif x <= 6.4e+19:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	else:
		tmp = (x * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.15e-14)
		tmp = Float64(Float64(log(y) / Float64(1.0 / Float64(x + -1.0))) - t);
	elseif (x <= 6.4e+19)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.15e-14)
		tmp = (log(y) / (1.0 / (x + -1.0))) - t;
	elseif (x <= 6.4e+19)
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	else
		tmp = (x * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.15e-14], N[(N[(N[Log[y], $MachinePrecision] / N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 6.4e+19], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.14999999999999999e-14

   1. Initial program 97.4%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \color{blue}{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), t\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), t\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), t\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), t\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), t\right) \]

      7. +-lowering-+.f64 97.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), t\right) \]

   5. Simplified 97.1%

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

      1. /-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\log y}{1} \cdot \left(-1 + x\right)\right), t\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\log y}{1} \cdot \left(x + -1\right)\right), t\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\log y}{\frac{1}{x + -1}}\right), t\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\log y, \left(\frac{1}{x + -1}\right)\right), t\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(y\right), \left(\frac{1}{x + -1}\right)\right), t\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{/.f64}\left(1, \left(x + -1\right)\right)\right), t\right) \]

      7. +-lowering-+.f64 97.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(x, -1\right)\right)\right), t\right) \]

   7. Applied egg-rr 97.2%

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

   if -1.14999999999999999e-14 < x < 6.4e19

   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. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right), t\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z - 1\right)\right)\right), t\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + -1\right)\right)\right), t\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(-1 + z\right)\right)\right), t\right) \]

      15. +-lowering-+.f64 98.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, z\right)\right)\right), t\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \log y - y \cdot \left(z - 1\right)\right), \color{blue}{t}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \log y + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right) + -1 \cdot \log y\right), t\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right) + \left(\mathsf{neg}\left(\log y\right)\right)\right), t\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right) - \log y\right), t\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right), \log y\right), t\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)\right), \log y\right), t\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \left(-1 \cdot \left(z - 1\right)\right)\right), \log y\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(-1 \cdot \left(z - 1\right)\right)\right), \log y\right), t\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)\right), \log y\right), t\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(0 - \left(z - 1\right)\right)\right), \log y\right), t\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\left(0 - z\right) + 1\right)\right), \log y\right), t\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(z\right)\right) + 1\right)\right), \log y\right), t\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), \log y\right), t\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(1 - z\right)\right), \log y\right), t\right) \]

      18. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, z\right)\right), \log y\right), t\right) \]

      19. log-lowering-log.f64 98.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{log.f64}\left(y\right)\right), t\right) \]

   8. Simplified 98.5%

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

   if 6.4e19 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, t\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot x\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, x\right), t\right) \]

      3. log-lowering-log.f64 98.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), x\right), t\right) \]

   5. Simplified 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-14}:\\ \;\;\;\;\frac{\log y}{\frac{1}{x + -1}} - t\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+19}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\left(y + \log y \cdot \left(x + -1\right)\right) - t\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+19}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4e-15)
   (- (+ y (* (log y) (+ x -1.0))) t)
   (if (<= x 6.4e+19) (- (- (* y (- 1.0 z)) (log y)) t) (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4e-15) {
		tmp = (y + (log(y) * (x + -1.0))) - t;
	} else if (x <= 6.4e+19) {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	} else {
		tmp = (x * 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 <= (-4d-15)) then
        tmp = (y + (log(y) * (x + (-1.0d0)))) - t
    else if (x <= 6.4d+19) then
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    else
        tmp = (x * 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 <= -4e-15) {
		tmp = (y + (Math.log(y) * (x + -1.0))) - t;
	} else if (x <= 6.4e+19) {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	} else {
		tmp = (x * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4e-15:
		tmp = (y + (math.log(y) * (x + -1.0))) - t
	elif x <= 6.4e+19:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	else:
		tmp = (x * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4e-15)
		tmp = Float64(Float64(y + Float64(log(y) * Float64(x + -1.0))) - t);
	elseif (x <= 6.4e+19)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4e-15)
		tmp = (y + (log(y) * (x + -1.0))) - t;
	elseif (x <= 6.4e+19)
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	else
		tmp = (x * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4e-15], N[(N[(y + N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 6.4e+19], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000003e-15

   1. Initial program 97.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right), t\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z - 1\right)\right)\right), t\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + -1\right)\right)\right), t\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(-1 + z\right)\right)\right), t\right) \]

      15. +-lowering-+.f64 99.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, z\right)\right)\right), t\right) \]

   5. Simplified 99.1%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\log y \cdot \left(x - 1\right) - -1 \cdot y\right)}, t\right) \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right), t\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + 1 \cdot y\right), t\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + y\right), t\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + \log y \cdot \left(x - 1\right)\right), t\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \left(\log y \cdot \left(x - 1\right)\right)\right), t\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\log y, \left(x - 1\right)\right)\right), t\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right)\right), t\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 96.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(x, -1\right)\right)\right), t\right) \]

   8. Simplified 96.6%

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

   if -4.0000000000000003e-15 < x < 6.4e19

   1. Initial program 86.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right), t\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z - 1\right)\right)\right), t\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + -1\right)\right)\right), t\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(-1 + z\right)\right)\right), t\right) \]

      15. +-lowering-+.f64 99.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, z\right)\right)\right), t\right) \]

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \log y - y \cdot \left(z - 1\right)\right), \color{blue}{t}\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \log y + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right) + -1 \cdot \log y\right), t\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right) + \left(\mathsf{neg}\left(\log y\right)\right)\right), t\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right) - \log y\right), t\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right), \log y\right), t\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)\right), \log y\right), t\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(y \cdot \left(-1 \cdot \left(z - 1\right)\right)\right), \log y\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(-1 \cdot \left(z - 1\right)\right)\right), \log y\right), t\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)\right), \log y\right), t\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(0 - \left(z - 1\right)\right)\right), \log y\right), t\right) \]

      14. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\left(0 - z\right) + 1\right)\right), \log y\right), t\right) \]

      15. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(z\right)\right) + 1\right)\right), \log y\right), t\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), \log y\right), t\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(1 - z\right)\right), \log y\right), t\right) \]

      18. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, z\right)\right), \log y\right), t\right) \]

      19. log-lowering-log.f64 98.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, z\right)\right), \mathsf{log.f64}\left(y\right)\right), t\right) \]

   8. Simplified 98.8%

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

   if 6.4e19 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, t\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot x\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, x\right), t\right) \]

      3. log-lowering-log.f64 98.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), x\right), t\right) \]

   5. Simplified 98.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-15}:\\ \;\;\;\;\left(y + \log y \cdot \left(x + -1\right)\right) - t\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+19}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(y \cdot \left(-1 \cdot \left(z - 1\right) + \frac{-1}{2} \cdot \left(y \cdot \left(z - 1\right)\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 6: 87.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;t \leq -240000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 550000000:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= t -240000000.0)
     t_1
     (if (<= t 550000000.0) (* (log y) (+ x -1.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (t <= -240000000.0) {
		tmp = t_1;
	} else if (t <= 550000000.0) {
		tmp = log(y) * (x + -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - t
    if (t <= (-240000000.0d0)) then
        tmp = t_1
    else if (t <= 550000000.0d0) then
        tmp = log(y) * (x + (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if (t <= -240000000.0) {
		tmp = t_1;
	} else if (t <= 550000000.0) {
		tmp = Math.log(y) * (x + -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if t <= -240000000.0:
		tmp = t_1
	elif t <= 550000000.0:
		tmp = math.log(y) * (x + -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (t <= -240000000.0)
		tmp = t_1;
	elseif (t <= 550000000.0)
		tmp = Float64(log(y) * Float64(x + -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - t;
	tmp = 0.0;
	if (t <= -240000000.0)
		tmp = t_1;
	elseif (t <= 550000000.0)
		tmp = log(y) * (x + -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t, -240000000.0], t$95$1, If[LessEqual[t, 550000000.0], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4e8 or 5.5e8 < t

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y\right)}, t\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot x\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, x\right), t\right) \]

      3. log-lowering-log.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), x\right), t\right) \]

   5. Simplified 96.4%

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

   if -2.4e8 < t < 5.5e8

   1. Initial program 87.7%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \color{blue}{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), t\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), t\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), t\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), t\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), t\right) \]

      7. +-lowering-+.f64 85.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), t\right) \]

   5. Simplified 85.7%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\log y, \color{blue}{\left(x - 1\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{x} - 1\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + \color{blue}{x}\right)\right) \]

      6. +-lowering-+.f64 84.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, \color{blue}{x}\right)\right) \]

   8. Simplified 84.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -240000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq 550000000:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right)\right) - t\\ \mathbf{if}\;t \leq -1.32 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+57}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1
         (- (* y (* z (+ -1.0 (* y (+ -0.5 (* y -0.3333333333333333)))))) t)))
   (if (<= t -1.32e+14) t_1 (if (<= t 5.1e+57) (* (log y) (+ x -1.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * (z * (-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))))) - t;
	double tmp;
	if (t <= -1.32e+14) {
		tmp = t_1;
	} else if (t <= 5.1e+57) {
		tmp = log(y) * (x + -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y * (z * ((-1.0d0) + (y * ((-0.5d0) + (y * (-0.3333333333333333d0))))))) - t
    if (t <= (-1.32d+14)) then
        tmp = t_1
    else if (t <= 5.1d+57) then
        tmp = log(y) * (x + (-1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * (z * (-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))))) - t;
	double tmp;
	if (t <= -1.32e+14) {
		tmp = t_1;
	} else if (t <= 5.1e+57) {
		tmp = Math.log(y) * (x + -1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * (z * (-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))))) - t
	tmp = 0
	if t <= -1.32e+14:
		tmp = t_1
	elif t <= 5.1e+57:
		tmp = math.log(y) * (x + -1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(-0.5 + Float64(y * -0.3333333333333333)))))) - t)
	tmp = 0.0
	if (t <= -1.32e+14)
		tmp = t_1;
	elseif (t <= 5.1e+57)
		tmp = Float64(log(y) * Float64(x + -1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * (z * (-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))))) - t;
	tmp = 0.0;
	if (t <= -1.32e+14)
		tmp = t_1;
	elseif (t <= 5.1e+57)
		tmp = log(y) * (x + -1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t, -1.32e+14], t$95$1, If[LessEqual[t, 5.1e+57], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.32e14 or 5.10000000000000023e57 < t

   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. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right)\right), t\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)\right), t\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), t\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right)\right), t\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(-1 + y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{2} + \frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right)\right), t\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(y \cdot \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(y \cdot \left(z \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)}, t\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right), t\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(-1 + y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right), t\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), t\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot y\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 78.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{-1}{3}\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

   8. Simplified 78.4%

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

   if -1.32e14 < t < 5.10000000000000023e57

   1. Initial program 88.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \color{blue}{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), t\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), t\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), t\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), t\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), t\right) \]

      7. +-lowering-+.f64 86.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), t\right) \]

   5. Simplified 86.0%

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

   6. Taylor expanded in t around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\log y, \color{blue}{\left(x - 1\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(\color{blue}{x} - 1\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + \color{blue}{x}\right)\right) \]

      6. +-lowering-+.f64 83.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, \color{blue}{x}\right)\right) \]

   8. Simplified 83.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right)\right) - t\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+57}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+82}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -8.5e+138)
     t_1
     (if (<= x 2.35e+82)
       (- (* y (* z (+ -1.0 (* y (+ -0.5 (* y -0.3333333333333333)))))) t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -8.5e+138) {
		tmp = t_1;
	} else if (x <= 2.35e+82) {
		tmp = (y * (z * (-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-8.5d+138)) then
        tmp = t_1
    else if (x <= 2.35d+82) then
        tmp = (y * (z * ((-1.0d0) + (y * ((-0.5d0) + (y * (-0.3333333333333333d0))))))) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -8.5e+138) {
		tmp = t_1;
	} else if (x <= 2.35e+82) {
		tmp = (y * (z * (-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -8.5e+138:
		tmp = t_1
	elif x <= 2.35e+82:
		tmp = (y * (z * (-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))))) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -8.5e+138)
		tmp = t_1;
	elseif (x <= 2.35e+82)
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(-0.5 + Float64(y * -0.3333333333333333)))))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -8.5e+138)
		tmp = t_1;
	elseif (x <= 2.35e+82)
		tmp = (y * (z * (-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))))) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e+138], t$95$1, If[LessEqual[x, 2.35e+82], N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000006e138 or 2.35e82 < x

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\log y, \color{blue}{x}\right) \]

      3. log-lowering-log.f64 81.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), x\right) \]

   5. Simplified 81.7%

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

   if -8.5000000000000006e138 < x < 2.35e82

   1. Initial program 88.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right)\right), t\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)\right), t\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), t\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right)\right), t\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(-1 + y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{2} + \frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right)\right), t\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(y \cdot \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 99.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(y \cdot \left(z \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)}, t\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right), t\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(-1 + y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right), t\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), t\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot y\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

      12. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{-1}{3}\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

   8. Simplified 53.6%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 + -0.5\right)\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+82}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right), t\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z - 1\right)\right)\right), t\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + -1\right)\right)\right), t\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(-1 + z\right)\right)\right), t\right) \]

   15. +-lowering-+.f64 99.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, z\right)\right)\right), t\right) \]

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 10: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right), t\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z - 1\right)\right)\right), t\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + -1\right)\right)\right), t\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(-1 + z\right)\right)\right), t\right) \]

   15. +-lowering-+.f64 99.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, z\right)\right)\right), t\right) \]

5. Simplified 99.2%

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

6. Taylor expanded in z around inf

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right), t\right) \]
7. Step-by-step derivation

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 11: 88.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right), t\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z - 1\right)\right)\right), t\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + -1\right)\right)\right), t\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(-1 + z\right)\right)\right), t\right) \]

   15. +-lowering-+.f64 99.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, z\right)\right)\right), t\right) \]

5. Simplified 99.2%

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

6. Taylor expanded in z around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\log y \cdot \left(x - 1\right) - -1 \cdot y\right)}, t\right) \]
7. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right), t\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + 1 \cdot y\right), t\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + y\right), t\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y + \log y \cdot \left(x - 1\right)\right), t\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \left(\log y \cdot \left(x - 1\right)\right)\right), t\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\log y, \left(x - 1\right)\right)\right), t\right) \]

   7. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right)\right), t\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 91.1%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(x, -1\right)\right)\right), t\right) \]

8. Simplified 91.1%

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

Alternative 12: 87.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in y around 0

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \color{blue}{t}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), t\right) \]

   3. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), t\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), t\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), t\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), t\right) \]

   7. +-lowering-+.f64 90.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), t\right) \]

5. Simplified 90.8%

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

6. Final simplification 90.8%

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

Alternative 13: 42.6% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00065:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -0.00065) (- 0.0 t) (if (<= t 7e+39) (* y (- 1.0 z)) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.00065) {
		tmp = 0.0 - t;
	} else if (t <= 7e+39) {
		tmp = y * (1.0 - z);
	} else {
		tmp = 0.0 - 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 <= (-0.00065d0)) then
        tmp = 0.0d0 - t
    else if (t <= 7d+39) then
        tmp = y * (1.0d0 - z)
    else
        tmp = 0.0d0 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.00065) {
		tmp = 0.0 - t;
	} else if (t <= 7e+39) {
		tmp = y * (1.0 - z);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -0.00065:
		tmp = 0.0 - t
	elif t <= 7e+39:
		tmp = y * (1.0 - z)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -0.00065)
		tmp = Float64(0.0 - t);
	elseif (t <= 7e+39)
		tmp = Float64(y * Float64(1.0 - z));
	else
		tmp = Float64(0.0 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -0.00065)
		tmp = 0.0 - t;
	elseif (t <= 7e+39)
		tmp = y * (1.0 - z);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -0.00065], N[(0.0 - t), $MachinePrecision], If[LessEqual[t, 7e+39], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.4999999999999997e-4 or 7.0000000000000003e39 < t

   1. Initial program 97.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{t} \]

      3. --lowering--.f64 72.9%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

   5. Simplified 72.9%

      \[\leadsto \color{blue}{0 - t} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 72.9%

         \[\leadsto \mathsf{neg.f64}\left(t\right) \]

   7. Applied egg-rr 72.9%

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

   if -6.4999999999999997e-4 < t < 7.0000000000000003e39

   1. Initial program 87.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right), t\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z - 1\right)\right)\right), t\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + -1\right)\right)\right), t\right) \]

      14. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(-1 + z\right)\right)\right), t\right) \]

      15. +-lowering-+.f64 98.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, z\right)\right)\right), t\right) \]

   5. Simplified 98.8%

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

   6. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

         \[\leadsto y \cdot \left(\left(0 - z\right) + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto y \cdot \left(0 - \color{blue}{\left(z - 1\right)}\right) \]

      5. neg-sub0 N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right) \]

      6. mul-1-neg N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(-1 \cdot \left(z - 1\right)\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(0 - \color{blue}{\left(z - 1\right)}\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(0 - z\right) + \color{blue}{1}\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(z\right)\right) + 1\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(1 - \color{blue}{z}\right)\right) \]

      14. --lowering--.f64 15.2%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{z}\right)\right) \]

   8. Simplified 15.2%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00065:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+39}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.3% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right)\right), t\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)\right), t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right)\right), t\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \left(-1 + y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{2} + \frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right)\right), t\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(y \cdot \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

   12. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

5. Simplified 99.7%

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

6. Taylor expanded in z around inf

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(y \cdot \left(z \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)}, t\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(z \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right), t\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(-1 + y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right), t\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right), t\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot y\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

   12. *-lowering-*.f64 42.6%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{-1}{3}\right), \frac{-1}{2}\right)\right)\right)\right)\right), t\right) \]

8. Simplified 42.6%

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

9. Final simplification 42.6%

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

Alternative 15: 46.1% 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 92.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)}, t\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + -1 \cdot \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) + \left(\mathsf{neg}\left(y \cdot \left(z - 1\right)\right)\right)\right), t\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right) - y \cdot \left(z - 1\right)\right), t\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\log y \cdot \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log y, \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   6. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x - 1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(x + -1\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \left(-1 + x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \left(y \cdot \left(z - 1\right)\right)\right), t\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z - 1\right)\right)\right), t\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(z + -1\right)\right)\right), t\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \left(-1 + z\right)\right)\right), t\right) \]

   15. +-lowering-+.f64 99.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \mathsf{+.f64}\left(-1, x\right)\right), \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, z\right)\right)\right), t\right) \]

5. Simplified 99.2%

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

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(y \cdot \left(1 - z\right)\right)}, t\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), t\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(\left(\mathsf{neg}\left(z\right)\right) + 1\right)\right), t\right) \]

   3. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(\left(0 - z\right) + 1\right)\right), t\right) \]

   4. associate-+l- N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(0 - \left(z - 1\right)\right)\right), t\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)\right), t\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(-1 \cdot \left(z - 1\right)\right)\right), t\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(-1 \cdot \left(z - 1\right)\right)\right), t\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right)\right), t\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(0 - \left(z - 1\right)\right)\right), t\right) \]

   10. associate-+l- N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\left(0 - z\right) + 1\right)\right), t\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(z\right)\right) + 1\right)\right), t\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)\right), t\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(1 - z\right)\right), t\right) \]

   14. --lowering--.f64 42.6%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, z\right)\right), t\right) \]

8. Simplified 42.6%

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

Alternative 16: 35.4% accurate, 71.7× speedup

Math

\[\begin{array}{l} \\ 0 - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 t))

C

double code(double x, double y, double z, double t) {
	return 0.0 - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0 - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - t;
}

Python

def code(x, y, z, t):
	return 0.0 - t

Julia

function code(x, y, z, t)
	return Float64(0.0 - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - t), $MachinePrecision]

Derivation

1. Initial program 92.1%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{t} \]

   3. --lowering--.f64 34.8%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

5. Simplified 34.8%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 34.8%

      \[\leadsto \mathsf{neg.f64}\left(t\right) \]

7. Applied egg-rr 34.8%

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

8. Final simplification 34.8%

   \[\leadsto 0 - t \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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))