Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Percentage Accurate: 86.0% → 99.8%

Time: 13.0s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

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

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

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

   4. neg-sub0 N/A

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

   5. --lowering--.f64 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.5 + y \cdot \left(-0.3333333333333333 + y \cdot -0.25\right)\\ \left(x \cdot \log y + \frac{\left(-1 + \left(y \cdot \left(y \cdot y\right)\right) \cdot \left(t\_1 \cdot \left(t\_1 \cdot t\_1\right)\right)\right) \cdot \left(y \cdot z\right)}{1 + \left(y \cdot t\_1\right) \cdot \left(y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right) - -1\right)}\right) - t \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ -0.5 (* y (+ -0.3333333333333333 (* y -0.25))))))
   (-
    (+
     (* x (log y))
     (/
      (* (+ -1.0 (* (* y (* y y)) (* t_1 (* t_1 t_1)))) (* y z))
      (+ 1.0 (* (* y t_1) (- (* y (+ -0.5 (* y -0.3333333333333333))) -1.0)))))
    t)))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 + (y * (-0.3333333333333333 + (y * -0.25)));
	return ((x * log(y)) + (((-1.0 + ((y * (y * y)) * (t_1 * (t_1 * t_1)))) * (y * z)) / (1.0 + ((y * t_1) * ((y * (-0.5 + (y * -0.3333333333333333))) - -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
    real(8) :: t_1
    t_1 = (-0.5d0) + (y * ((-0.3333333333333333d0) + (y * (-0.25d0))))
    code = ((x * log(y)) + ((((-1.0d0) + ((y * (y * y)) * (t_1 * (t_1 * t_1)))) * (y * z)) / (1.0d0 + ((y * t_1) * ((y * ((-0.5d0) + (y * (-0.3333333333333333d0)))) - (-1.0d0)))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 + (y * (-0.3333333333333333 + (y * -0.25)));
	return ((x * Math.log(y)) + (((-1.0 + ((y * (y * y)) * (t_1 * (t_1 * t_1)))) * (y * z)) / (1.0 + ((y * t_1) * ((y * (-0.5 + (y * -0.3333333333333333))) - -1.0))))) - t;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 + (y * (-0.3333333333333333 + (y * -0.25)))
	return ((x * math.log(y)) + (((-1.0 + ((y * (y * y)) * (t_1 * (t_1 * t_1)))) * (y * z)) / (1.0 + ((y * t_1) * ((y * (-0.5 + (y * -0.3333333333333333))) - -1.0))))) - t

Julia

function code(x, y, z, t)
	t_1 = Float64(-0.5 + Float64(y * Float64(-0.3333333333333333 + Float64(y * -0.25))))
	return Float64(Float64(Float64(x * log(y)) + Float64(Float64(Float64(-1.0 + Float64(Float64(y * Float64(y * y)) * Float64(t_1 * Float64(t_1 * t_1)))) * Float64(y * z)) / Float64(1.0 + Float64(Float64(y * t_1) * Float64(Float64(y * Float64(-0.5 + Float64(y * -0.3333333333333333))) - -1.0))))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = -0.5 + (y * (-0.3333333333333333 + (y * -0.25)));
	tmp = ((x * log(y)) + (((-1.0 + ((y * (y * y)) * (t_1 * (t_1 * t_1)))) * (y * z)) / (1.0 + ((y * t_1) * ((y * (-0.5 + (y * -0.3333333333333333))) - -1.0))))) - t;
end

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in y around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

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

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

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

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

   16. *-commutative N/A

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

   17. *-lowering-*.f64 99.6%

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

5. Simplified 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 99.6%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, \frac{-1}{4}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, \frac{-1}{4}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, \frac{-1}{4}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right), -1\right)\right)\right)\right)\right), t\right) \]

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in y around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

5. Simplified 99.6%

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

Alternative 4: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in y around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

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

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

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

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

   16. *-commutative N/A

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

   17. *-lowering-*.f64 99.6%

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

5. Simplified 99.6%

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

Alternative 5: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

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

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

   5. mul-1-neg N/A

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

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

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in y around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 99.5%

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

5. Simplified 99.5%

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

Alternative 7: 90.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -4 \cdot 10^{-118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-45}:\\ \;\;\;\;y \cdot \left(0 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -4e-118) t_1 (if (<= x 2.3e-45) (- (* y (- 0.0 z)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -4e-118) {
		tmp = t_1;
	} else if (x <= 2.3e-45) {
		tmp = (y * (0.0 - z)) - 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)) - t
    if (x <= (-4d-118)) then
        tmp = t_1
    else if (x <= 2.3d-45) then
        tmp = (y * (0.0d0 - z)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -4e-118) {
		tmp = t_1;
	} else if (x <= 2.3e-45) {
		tmp = (y * (0.0 - z)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -4e-118:
		tmp = t_1
	elif x <= 2.3e-45:
		tmp = (y * (0.0 - z)) - t
	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 (x <= -4e-118)
		tmp = t_1;
	elseif (x <= 2.3e-45)
		tmp = Float64(Float64(y * Float64(0.0 - z)) - t);
	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 (x <= -4e-118)
		tmp = t_1;
	elseif (x <= 2.3e-45)
		tmp = (y * (0.0 - z)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -4e-118], t$95$1, If[LessEqual[x, 2.3e-45], N[(N[(y * N[(0.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.99999999999999994e-118 or 2.29999999999999992e-45 < x

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

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

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

      4. mul-1-neg N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. log-rec N/A

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

      9. remove-double-neg N/A

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

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

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

      11. log-lowering-log.f64 92.3%

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

   5. Simplified 92.3%

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

   if -3.99999999999999994e-118 < x < 2.29999999999999992e-45

   1. Initial program 68.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. log-rec N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 94.0%

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

   8. Simplified 94.0%

      \[\leadsto \color{blue}{\left(0 - z \cdot y\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-45}:\\ \;\;\;\;y \cdot \left(0 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

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

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

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

   4. neg-sub0 N/A

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

   5. --lowering--.f64 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt-out N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. +-commutative N/A

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 99.2%

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

7. Simplified 99.2%

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

Alternative 9: 75.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{+175}:\\ \;\;\;\;y \cdot \left(0 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.5e+79) t_1 (if (<= x 1e+175) (- (* y (- 0.0 z)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.5e+79) {
		tmp = t_1;
	} else if (x <= 1e+175) {
		tmp = (y * (0.0 - z)) - 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 <= (-1.5d+79)) then
        tmp = t_1
    else if (x <= 1d+175) then
        tmp = (y * (0.0d0 - z)) - 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 <= -1.5e+79) {
		tmp = t_1;
	} else if (x <= 1e+175) {
		tmp = (y * (0.0 - z)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -1.5e+79:
		tmp = t_1
	elif x <= 1e+175:
		tmp = (y * (0.0 - z)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.5e+79)
		tmp = t_1;
	elseif (x <= 1e+175)
		tmp = Float64(Float64(y * Float64(0.0 - z)) - 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 <= -1.5e+79)
		tmp = t_1;
	elseif (x <= 1e+175)
		tmp = (y * (0.0 - z)) - 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, -1.5e+79], t$95$1, If[LessEqual[x, 1e+175], N[(N[(y * N[(0.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999987e79 or 9.9999999999999994e174 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 78.3%

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

   5. Simplified 78.3%

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

   if -1.49999999999999987e79 < x < 9.9999999999999994e174

   1. Initial program 78.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. log-rec N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 80.3%

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

   8. Simplified 80.3%

      \[\leadsto \color{blue}{\left(0 - z \cdot y\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 10^{+175}:\\ \;\;\;\;y \cdot \left(0 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. log-rec N/A

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

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

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

   7. mul-1-neg N/A

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

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

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

   9. mul-1-neg N/A

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

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

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

   11. log-rec N/A

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

   12. remove-double-neg N/A

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 98.3%

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

5. Simplified 98.3%

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

6. Final simplification 98.3%

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

Alternative 11: 48.9% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.75 \cdot 10^{-86}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.75e-86) (- 0.0 t) (if (<= t 5.2e-93) (* y (- 0.0 z)) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.75e-86) {
		tmp = 0.0 - t;
	} else if (t <= 5.2e-93) {
		tmp = y * (0.0 - 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 <= (-2.75d-86)) then
        tmp = 0.0d0 - t
    else if (t <= 5.2d-93) then
        tmp = y * (0.0d0 - 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 <= -2.75e-86) {
		tmp = 0.0 - t;
	} else if (t <= 5.2e-93) {
		tmp = y * (0.0 - z);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.75e-86:
		tmp = 0.0 - t
	elif t <= 5.2e-93:
		tmp = y * (0.0 - z)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.75e-86)
		tmp = Float64(0.0 - t);
	elseif (t <= 5.2e-93)
		tmp = Float64(y * Float64(0.0 - 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 <= -2.75e-86)
		tmp = 0.0 - t;
	elseif (t <= 5.2e-93)
		tmp = y * (0.0 - z);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.75e-86], N[(0.0 - t), $MachinePrecision], If[LessEqual[t, 5.2e-93], N[(y * N[(0.0 - z), $MachinePrecision]), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.75e-86 or 5.1999999999999997e-93 < t

   1. Initial program 91.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 68.9%

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

   5. Simplified 68.9%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 68.9%

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

   7. Applied egg-rr 68.9%

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

   if -2.75e-86 < t < 5.1999999999999997e-93

   1. Initial program 67.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. log-rec N/A

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

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

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

      7. mul-1-neg N/A

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

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

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

      9. mul-1-neg N/A

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

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

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

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

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{y \cdot z} \]

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 37.8%

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

   8. Simplified 37.8%

      \[\leadsto \color{blue}{0 - z \cdot y} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

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

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

      5. neg-lowering-neg.f64 37.8%

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

   10. Applied egg-rr 37.8%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.75 \cdot 10^{-86}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-93}:\\ \;\;\;\;y \cdot \left(0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.9% accurate, 30.1× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(0 - z\right) - t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (y * (0.0 - 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 * (0.0d0 - z)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. log-rec N/A

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

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

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

   7. mul-1-neg N/A

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

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

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

   9. mul-1-neg N/A

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

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

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

   11. log-rec N/A

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

   12. remove-double-neg N/A

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

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

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

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

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

   15. *-commutative N/A

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

   16. *-lowering-*.f64 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 65.1%

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

8. Simplified 65.1%

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

9. Final simplification 65.1%

   \[\leadsto y \cdot \left(0 - z\right) - t \]
10. Add Preprocessing

Alternative 13: 43.1% accurate, 70.3× 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 83.7%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 47.9%

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

5. Simplified 47.9%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 47.9%

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

7. Applied egg-rr 47.9%

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

8. Final simplification 47.9%

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

Developer Target 1: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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 * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))