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

Percentage Accurate: 89.4% → 99.8%

Time: 16.8s

Alternatives: 22

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. sub-neg 89.8%

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

   2. +-commutative 89.8%

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

   3. associate-+l+ 89.8%

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

   4. fma-define 89.8%

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

   5. sub-neg 89.8%

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

   6. metadata-eval 89.8%

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

   7. sub-neg 89.8%

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

   8. log1p-define 99.8%

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

   9. fma-define 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. fma-define 89.8%

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

   2. sub-neg 89.8%

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

   3. metadata-eval 89.8%

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

   4. sub-neg 89.8%

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

   5. metadata-eval 89.8%

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

   6. sub-neg 89.8%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. fma-define 89.8%

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

   2. sub-neg 89.8%

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

   3. metadata-eval 89.8%

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

   4. sub-neg 89.8%

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

   5. metadata-eval 89.8%

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

   6. sub-neg 89.8%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around inf 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in z around inf 89.5%

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

   1. *-commutative 89.5%

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

   2. sub-neg 89.5%

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

   3. log1p-define 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 99.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((((-1.0d0) + x) * log(y)) + (y * ((1.0d0 - z) + (y * (((z + (-1.0d0)) * (-0.5d0)) + (y * (((z + (-1.0d0)) * (-0.3333333333333333d0)) + ((-0.25d0) * (y * (z + (-1.0d0))))))))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in y around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 6: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((((-1.0d0) + x) * log(y)) + ((z + (-1.0d0)) * (y * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in y around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 7: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -2 \lor \neg \left(-1 + x \leq -1\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - \left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -2.0) (not (<= (+ -1.0 x) -1.0)))
   (- (* (+ -1.0 x) (log y)) t)
   (- (* y (- 1.0 z)) (+ (log y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -2.0) || !((-1.0 + x) <= -1.0)) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = (y * (1.0 - z)) - (log(y) + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-2.0d0)) .or. (.not. (((-1.0d0) + x) <= (-1.0d0)))) then
        tmp = (((-1.0d0) + x) * log(y)) - t
    else
        tmp = (y * (1.0d0 - z)) - (log(y) + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -2.0) || !((-1.0 + x) <= -1.0)) {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	} else {
		tmp = (y * (1.0 - z)) - (Math.log(y) + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -2.0) or not ((-1.0 + x) <= -1.0):
		tmp = ((-1.0 + x) * math.log(y)) - t
	else:
		tmp = (y * (1.0 - z)) - (math.log(y) + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -2.0) || !(Float64(-1.0 + x) <= -1.0))
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(1.0 - z)) - Float64(log(y) + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -2.0) || ~(((-1.0 + x) <= -1.0)))
		tmp = ((-1.0 + x) * log(y)) - t;
	else
		tmp = (y * (1.0 - z)) - (log(y) + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -2.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0]], $MachinePrecision]], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.9%

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

      1. fma-define 95.9%

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

      2. sub-neg 95.9%

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

      3. metadata-eval 95.9%

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

      4. sub-neg 95.9%

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

      5. metadata-eval 95.9%

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

      6. sub-neg 95.9%

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

      7. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.6%

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

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

   1. Initial program 82.9%

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

      1. sub-neg 82.9%

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

      2. +-commutative 82.9%

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

      3. associate-+l+ 82.9%

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

      4. fma-define 82.9%

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

      5. sub-neg 82.9%

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

      6. metadata-eval 82.9%

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

      7. sub-neg 82.9%

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

      8. log1p-define 99.9%

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

      9. fma-define 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 98.0%

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

      1. associate-*r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. metadata-eval 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

   10. Simplified 98.0%

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

4. Final simplification 96.2%

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

Alternative 8: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in y around 0 99.2%

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

4. Final simplification 99.2%

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

Alternative 9: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in y around 0 99.1%

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

4. Final simplification 99.1%

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

Alternative 10: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+62} \lor \neg \left(x \leq 3.9 \cdot 10^{+39}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(y - t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -9.5e+62) (not (<= x 3.9e+39)))
   (* x (log y))
   (- (- y t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -9.5e+62) || !(x <= 3.9e+39)) {
		tmp = x * log(y);
	} else {
		tmp = (y - t) - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-9.5d+62)) .or. (.not. (x <= 3.9d+39))) then
        tmp = x * log(y)
    else
        tmp = (y - t) - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -9.5e+62) || !(x <= 3.9e+39)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y - t) - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -9.5e+62) or not (x <= 3.9e+39):
		tmp = x * math.log(y)
	else:
		tmp = (y - t) - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -9.5e+62) || !(x <= 3.9e+39))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y - t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -9.5e+62) || ~((x <= 3.9e+39)))
		tmp = x * log(y);
	else
		tmp = (y - t) - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -9.5e+62], N[Not[LessEqual[x, 3.9e+39]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y - t), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000003e62 or 3.9000000000000001e39 < x

   1. Initial program 97.0%

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

      1. fma-define 97.0%

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

      2. sub-neg 97.0%

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

      3. metadata-eval 97.0%

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

      4. sub-neg 97.0%

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

      5. metadata-eval 97.0%

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

      6. sub-neg 97.0%

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

      7. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in x around inf 76.1%

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

      1. *-commutative 76.1%

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

   8. Simplified 76.1%

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

   if -9.5000000000000003e62 < x < 3.9000000000000001e39

   1. Initial program 84.3%

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

      1. sub-neg 84.3%

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

      2. +-commutative 84.3%

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

      3. associate-+l+ 84.3%

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

      4. fma-define 84.3%

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

      5. sub-neg 84.3%

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

      6. metadata-eval 84.3%

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

      7. sub-neg 84.3%

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

      8. log1p-define 99.9%

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

      9. fma-define 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 95.1%

      \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 95.1%

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

   7. Simplified 95.1%

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

   8. Taylor expanded in y around 0 93.0%

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

      1. associate-*r* 93.0%

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

      2. mul-1-neg 93.0%

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

      3. sub-neg 93.0%

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

      4. metadata-eval 93.0%

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

      5. +-commutative 93.0%

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

      6. +-commutative 93.0%

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

   10. Simplified 93.0%

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

   11. Taylor expanded in z around 0 76.8%

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

       1. associate--r+ 76.8%

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

   13. Simplified 76.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+62} \lor \neg \left(x \leq 3.9 \cdot 10^{+39}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(y - t\right) - \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+67} \lor \neg \left(x \leq 1.22 \cdot 10^{+42}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7e+67) (not (<= x 1.22e+42))) (* x (log y)) (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7e+67) || !(x <= 1.22e+42)) {
		tmp = x * log(y);
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-7d+67)) .or. (.not. (x <= 1.22d+42))) then
        tmp = x * log(y)
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7e+67) || !(x <= 1.22e+42)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -7e+67) or not (x <= 1.22e+42):
		tmp = x * math.log(y)
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7e+67) || !(x <= 1.22e+42))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -7e+67) || ~((x <= 1.22e+42)))
		tmp = x * log(y);
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7e+67], N[Not[LessEqual[x, 1.22e+42]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7e67 or 1.22e42 < x

   1. Initial program 97.0%

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

      1. fma-define 97.0%

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

      2. sub-neg 97.0%

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

      3. metadata-eval 97.0%

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

      4. sub-neg 97.0%

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

      5. metadata-eval 97.0%

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

      6. sub-neg 97.0%

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

      7. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in x around inf 76.1%

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

      1. *-commutative 76.1%

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

   8. Simplified 76.1%

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

   if -7e67 < x < 1.22e42

   1. Initial program 84.3%

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

      1. sub-neg 84.3%

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

      2. +-commutative 84.3%

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

      3. associate-+l+ 84.3%

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

      4. fma-define 84.3%

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

      5. sub-neg 84.3%

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

      6. metadata-eval 84.3%

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

      7. sub-neg 84.3%

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

      8. log1p-define 99.9%

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

      9. fma-define 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 95.1%

      \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 95.1%

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

   7. Simplified 95.1%

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

   8. Taylor expanded in y around 0 76.3%

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

      1. mul-1-neg 76.3%

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

      2. distribute-neg-in 76.3%

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

      3. sub-neg 76.3%

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

   10. Simplified 76.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+67} \lor \neg \left(x \leq 1.22 \cdot 10^{+42}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+64} \lor \neg \left(x \leq 7.8 \cdot 10^{+41}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.05e+64) (not (<= x 7.8e+41)))
   (* x (log y))
   (- (* y (- 1.0 z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.05e+64) || !(x <= 7.8e+41)) {
		tmp = x * log(y);
	} else {
		tmp = (y * (1.0 - z)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.05d+64)) .or. (.not. (x <= 7.8d+41))) then
        tmp = x * log(y)
    else
        tmp = (y * (1.0d0 - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.05e+64) || !(x <= 7.8e+41)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * (1.0 - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.05e+64) or not (x <= 7.8e+41):
		tmp = x * math.log(y)
	else:
		tmp = (y * (1.0 - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.05e+64) || !(x <= 7.8e+41))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(1.0 - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.05e+64) || ~((x <= 7.8e+41)))
		tmp = x * log(y);
	else
		tmp = (y * (1.0 - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.05e+64], N[Not[LessEqual[x, 7.8e+41]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e64 or 7.7999999999999994e41 < x

   1. Initial program 97.0%

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

      1. fma-define 97.0%

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

      2. sub-neg 97.0%

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

      3. metadata-eval 97.0%

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

      4. sub-neg 97.0%

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

      5. metadata-eval 97.0%

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

      6. sub-neg 97.0%

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

      7. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in x around inf 76.1%

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

      1. *-commutative 76.1%

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

   8. Simplified 76.1%

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

   if -1.05e64 < x < 7.7999999999999994e41

   1. Initial program 84.3%

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

      1. sub-neg 84.3%

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

      2. +-commutative 84.3%

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

      3. associate-+l+ 84.3%

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

      4. fma-define 84.3%

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

      5. sub-neg 84.3%

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

      6. metadata-eval 84.3%

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

      7. sub-neg 84.3%

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

      8. log1p-define 99.9%

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

      9. fma-define 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 95.1%

      \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 95.1%

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

   7. Simplified 95.1%

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

   8. Taylor expanded in y around 0 93.0%

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

      1. associate-*r* 93.0%

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

      2. mul-1-neg 93.0%

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

      3. sub-neg 93.0%

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

      4. metadata-eval 93.0%

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

      5. +-commutative 93.0%

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

      6. +-commutative 93.0%

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

   10. Simplified 93.0%

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

   11. Taylor expanded in t around inf 64.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+64} \lor \neg \left(x \leq 7.8 \cdot 10^{+41}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+245}:\\ \;\;\;\;z \cdot \left(\mathsf{log1p}\left(-y\right) - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.1e+245)
   (* z (- (log1p (- y)) (/ t z)))
   (- (* (+ -1.0 x) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.1e+245) {
		tmp = z * (log1p(-y) - (t / z));
	} else {
		tmp = ((-1.0 + x) * log(y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.1e+245) {
		tmp = z * (Math.log1p(-y) - (t / z));
	} else {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.1e+245:
		tmp = z * (math.log1p(-y) - (t / z))
	else:
		tmp = ((-1.0 + x) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.1e+245)
		tmp = Float64(z * Float64(log1p(Float64(-y)) - Float64(t / z)));
	else
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.1e+245], N[(z * N[(N[Log[1 + (-y)], $MachinePrecision] - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.09999999999999996e245

   1. Initial program 30.7%

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

      1. fma-define 30.7%

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

      2. sub-neg 30.7%

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

      3. metadata-eval 30.7%

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

      4. sub-neg 30.7%

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

      5. metadata-eval 30.7%

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

      6. sub-neg 30.7%

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

      7. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 99.9%

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

   6. Taylor expanded in z around inf 30.7%

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

      1. associate--l+ 30.7%

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

      2. div-sub 30.7%

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

      3. sub-neg 30.7%

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

      4. log1p-define 99.9%

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

      5. sub-neg 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in t around inf 85.8%

      \[\leadsto z \cdot \left(\mathsf{log1p}\left(-y\right) + \color{blue}{-1 \cdot \frac{t}{z}}\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 85.8%

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

       2. neg-mul-1 85.8%

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

   11. Simplified 85.8%

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

   if -2.09999999999999996e245 < z

   1. Initial program 93.2%

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

      1. fma-define 93.2%

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

      2. sub-neg 93.2%

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

      3. metadata-eval 93.2%

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

      4. sub-neg 93.2%

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

      5. metadata-eval 93.2%

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

      6. sub-neg 93.2%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 91.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+245}:\\ \;\;\;\;z \cdot \left(\mathsf{log1p}\left(-y\right) - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. fma-define 89.8%

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

   2. sub-neg 89.8%

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

   3. metadata-eval 89.8%

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

   4. sub-neg 89.8%

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

   5. metadata-eval 89.8%

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

   6. sub-neg 89.8%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around 0 98.5%

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

   1. +-commutative 98.5%

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

   2. sub-neg 98.5%

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

   3. metadata-eval 98.5%

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

   4. mul-1-neg 98.5%

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

   5. unsub-neg 98.5%

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

   6. +-commutative 98.5%

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

   7. sub-neg 98.5%

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

   8. metadata-eval 98.5%

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

   9. +-commutative 98.5%

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 15: 89.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+246}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.52e+246) (- (- t) (* z y)) (- (* (+ -1.0 x) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.52e+246) {
		tmp = -t - (z * y);
	} else {
		tmp = ((-1.0 + x) * log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.52d+246)) then
        tmp = -t - (z * y)
    else
        tmp = (((-1.0d0) + x) * log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.52e+246) {
		tmp = -t - (z * y);
	} else {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.52e+246:
		tmp = -t - (z * y)
	else:
		tmp = ((-1.0 + x) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.52e+246)
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.52e+246)
		tmp = -t - (z * y);
	else
		tmp = ((-1.0 + x) * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.52e+246], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.52e246

   1. Initial program 30.7%

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

      1. sub-neg 30.7%

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

      2. +-commutative 30.7%

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

      3. associate-+l+ 30.7%

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

      4. fma-define 30.7%

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

      5. sub-neg 30.7%

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

      6. metadata-eval 30.7%

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

      7. sub-neg 30.7%

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

      8. log1p-define 99.9%

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

      9. fma-define 99.9%

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

      10. sub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 85.8%

      \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 85.8%

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

   7. Simplified 85.8%

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

   8. Taylor expanded in y around 0 82.6%

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

      1. associate-*r* 82.6%

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

      2. mul-1-neg 82.6%

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

      3. sub-neg 82.6%

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

      4. metadata-eval 82.6%

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

      5. +-commutative 82.6%

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

      6. +-commutative 82.6%

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

   10. Simplified 82.6%

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

   11. Taylor expanded in t around inf 82.6%

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

   12. Taylor expanded in z around inf 82.6%

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

   if -1.52e246 < z

   1. Initial program 93.2%

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

      1. fma-define 93.2%

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

      2. sub-neg 93.2%

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

      3. metadata-eval 93.2%

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

      4. sub-neg 93.2%

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

      5. metadata-eval 93.2%

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

      6. sub-neg 93.2%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 91.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+246}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. fma-define 89.8%

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

   2. sub-neg 89.8%

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

   3. metadata-eval 89.8%

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

   4. sub-neg 89.8%

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

   5. metadata-eval 89.8%

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

   6. sub-neg 89.8%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around inf 99.5%

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

6. Taylor expanded in y around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. sub-neg 98.2%

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

   3. metadata-eval 98.2%

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

   4. mul-1-neg 98.2%

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

   5. unsub-neg 98.2%

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

   6. +-commutative 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 17: 43.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.42 \cdot 10^{-11} \lor \neg \left(t \leq 3100\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.42e-11) (not (<= t 3100.0))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.42e-11) || !(t <= 3100.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.42d-11)) .or. (.not. (t <= 3100.0d0))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.42e-11) || !(t <= 3100.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.42e-11) or not (t <= 3100.0):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.42e-11) || !(t <= 3100.0))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.42e-11) || ~((t <= 3100.0)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.42e-11], N[Not[LessEqual[t, 3100.0]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.42e-11 or 3100 < t

   1. Initial program 94.5%

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

      1. fma-define 94.5%

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

      2. sub-neg 94.5%

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

      3. metadata-eval 94.5%

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

      4. sub-neg 94.5%

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

      5. metadata-eval 94.5%

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

      6. sub-neg 94.5%

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

      7. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 70.9%

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

      1. mul-1-neg 70.9%

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

   7. Simplified 70.9%

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

   if -1.42e-11 < t < 3100

   1. Initial program 85.5%

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

      1. sub-neg 85.5%

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

      2. +-commutative 85.5%

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

      3. associate-+l+ 85.5%

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

      4. fma-define 85.5%

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

      5. sub-neg 85.5%

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

      6. metadata-eval 85.5%

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

      7. sub-neg 85.5%

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

      8. log1p-define 99.7%

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

      9. fma-define 99.7%

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

      10. sub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 48.9%

      \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in y around 0 47.6%

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

      1. associate-*r* 47.6%

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

      2. mul-1-neg 47.6%

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

      3. sub-neg 47.6%

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

      4. metadata-eval 47.6%

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

      5. +-commutative 47.6%

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

      6. +-commutative 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in t around inf 18.1%

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

   12. Taylor expanded in z around inf 17.6%

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

       1. neg-mul-1 17.6%

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

       2. distribute-rgt-neg-in 17.6%

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

   14. Simplified 17.6%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.42 \cdot 10^{-11} \lor \neg \left(t \leq 3100\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.2% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. sub-neg 89.8%

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

   2. +-commutative 89.8%

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

   3. associate-+l+ 89.8%

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

   4. fma-define 89.8%

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

   5. sub-neg 89.8%

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

   6. metadata-eval 89.8%

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

   7. sub-neg 89.8%

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

   8. log1p-define 99.8%

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

   9. fma-define 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 64.1%

   \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 64.1%

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

7. Simplified 64.1%

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

8. Taylor expanded in y around 0 62.8%

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

   1. associate-*r* 62.8%

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

   2. mul-1-neg 62.8%

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

   3. sub-neg 62.8%

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

   4. metadata-eval 62.8%

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

   5. +-commutative 62.8%

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

   6. +-commutative 62.8%

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

10. Simplified 62.8%

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

11. Taylor expanded in t around inf 46.6%

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

12. Final simplification 46.6%

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

Alternative 19: 46.0% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. sub-neg 89.8%

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

   2. +-commutative 89.8%

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

   3. associate-+l+ 89.8%

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

   4. fma-define 89.8%

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

   5. sub-neg 89.8%

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

   6. metadata-eval 89.8%

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

   7. sub-neg 89.8%

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

   8. log1p-define 99.8%

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

   9. fma-define 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 64.1%

   \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 64.1%

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

7. Simplified 64.1%

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

8. Taylor expanded in y around 0 62.8%

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

   1. associate-*r* 62.8%

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

   2. mul-1-neg 62.8%

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

   3. sub-neg 62.8%

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

   4. metadata-eval 62.8%

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

   5. +-commutative 62.8%

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

   6. +-commutative 62.8%

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

10. Simplified 62.8%

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

11. Taylor expanded in t around inf 46.6%

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

12. Taylor expanded in z around inf 46.4%

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

13. Final simplification 46.4%

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

Alternative 20: 36.1% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. sub-neg 89.8%

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

   2. +-commutative 89.8%

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

   3. associate-+l+ 89.8%

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

   4. fma-define 89.8%

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

   5. sub-neg 89.8%

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

   6. metadata-eval 89.8%

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

   7. sub-neg 89.8%

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

   8. log1p-define 99.8%

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

   9. fma-define 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 64.1%

   \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \color{blue}{-1 \cdot \log y - t}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 64.1%

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

7. Simplified 64.1%

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

8. Taylor expanded in y around 0 62.8%

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

   1. associate-*r* 62.8%

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

   2. mul-1-neg 62.8%

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

   3. sub-neg 62.8%

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

   4. metadata-eval 62.8%

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

   5. +-commutative 62.8%

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

   6. +-commutative 62.8%

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

10. Simplified 62.8%

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

11. Taylor expanded in t around inf 46.6%

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

12. Taylor expanded in z around 0 36.2%

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

Alternative 21: 35.8% accurate, 107.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. fma-define 89.8%

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

   2. sub-neg 89.8%

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

   3. metadata-eval 89.8%

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

   4. sub-neg 89.8%

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

   5. metadata-eval 89.8%

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

   6. sub-neg 89.8%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 35.8%

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

   1. mul-1-neg 35.8%

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

7. Simplified 35.8%

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

Alternative 22: 2.2% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 89.8%

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

   1. fma-define 89.8%

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

   2. sub-neg 89.8%

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

   3. metadata-eval 89.8%

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

   4. sub-neg 89.8%

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

   5. metadata-eval 89.8%

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

   6. sub-neg 89.8%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 35.8%

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

   1. mul-1-neg 35.8%

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

7. Simplified 35.8%

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

   1. add-sqr-sqrt 18.1%

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

   2. sqrt-unprod 11.0%

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

   3. sqr-neg 11.0%

      \[\leadsto \sqrt{\color{blue}{t \cdot t}} \]

   4. sqrt-unprod 1.3%

      \[\leadsto \color{blue}{\sqrt{t} \cdot \sqrt{t}} \]

   5. add-sqr-sqrt 2.1%

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

   6. *-un-lft-identity 2.1%

      \[\leadsto \color{blue}{1 \cdot t} \]

9. Applied egg-rr 2.1%

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

    1. *-lft-identity 2.1%

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

11. Simplified 2.1%

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

Reproduce

herbie shell --seed 2024144 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))