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

Percentage Accurate: 96.9% → 98.9%

Time: 13.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;y \leq 1.1 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= y 1.1e-47) t_1 (fma z (log1p (- y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (y <= 1.1e-47) {
		tmp = t_1;
	} else {
		tmp = fma(z, log1p(-y), t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (y <= 1.1e-47)
		tmp = t_1;
	else
		tmp = fma(z, log1p(Float64(-y)), t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[y, 1.1e-47], t$95$1, N[(z * N[Log[1 + (-y)], $MachinePrecision] + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.10000000000000009e-47

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.8%

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

   if 1.10000000000000009e-47 < y

   1. Initial program 74.7%

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

      1. +-commutative 74.7%

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

      2. associate--l+ 74.7%

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

      3. fma-def 74.7%

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

      4. sub-neg 74.7%

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

      5. log1p-def 94.4%

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

   3. Simplified 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-47}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Final simplification 95.7%

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

Alternative 3: 71.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+63} \lor \neg \left(t \leq 66000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5.8e+63) (not (<= t 66000000.0))) (- t) (* x (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.8e+63) || !(t <= 66000000.0)) {
		tmp = -t;
	} else {
		tmp = x * 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 ((t <= (-5.8d+63)) .or. (.not. (t <= 66000000.0d0))) then
        tmp = -t
    else
        tmp = x * log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.8e+63) || !(t <= 66000000.0)) {
		tmp = -t;
	} else {
		tmp = x * Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -5.8e+63) or not (t <= 66000000.0):
		tmp = -t
	else:
		tmp = x * math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5.8e+63) || !(t <= 66000000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(x * log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -5.8e+63) || ~((t <= 66000000.0)))
		tmp = -t;
	else
		tmp = x * log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5.8e+63], N[Not[LessEqual[t, 66000000.0]], $MachinePrecision]], (-t), N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.7999999999999999e63 or 6.6e7 < t

   1. Initial program 98.7%

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

   3. Taylor expanded in t around inf 81.7%

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

      1. mul-1-neg 81.7%

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

   5. Simplified 81.7%

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

   if -5.7999999999999999e63 < t < 6.6e7

   1. Initial program 93.0%

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

      1. sub-neg 93.0%

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

      2. add-sqr-sqrt 54.9%

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

      3. sqrt-unprod 83.9%

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

      4. sqr-neg 83.9%

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

      5. sqrt-unprod 31.1%

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

      6. add-sqr-sqrt 70.9%

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

      7. expm1-log1p-u 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + t\right)\right)} \]

      8. add-sqr-sqrt 16.9%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \color{blue}{\sqrt{t} \cdot \sqrt{t}}\right)\right) \]

      9. sqrt-unprod 58.6%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \color{blue}{\sqrt{t \cdot t}}\right)\right) \]

      10. sqr-neg 58.6%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \sqrt{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right)\right) \]

      11. sqrt-unprod 43.9%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \color{blue}{\sqrt{-t} \cdot \sqrt{-t}}\right)\right) \]

      12. add-sqr-sqrt 67.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \color{blue}{\left(-t\right)}\right)\right) \]

      13. sub-neg 67.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t}\right)\right) \]

      14. +-commutative 67.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(z \cdot \log \left(1 - y\right) + x \cdot \log y\right)} - t\right)\right) \]

      15. associate-+r- 67.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{z \cdot \log \left(1 - y\right) + \left(x \cdot \log y - t\right)}\right)\right) \]

      16. sub-neg 67.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(z \cdot \log \color{blue}{\left(1 + \left(-y\right)\right)} + \left(x \cdot \log y - t\right)\right)\right) \]

      17. log1p-udef 52.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} + \left(x \cdot \log y - t\right)\right)\right) \]

      18. fma-udef 52.8%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right)}\right)\right) \]

   4. Applied egg-rr 36.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(z, \mathsf{log1p}\left(y\right), \mathsf{fma}\left(x, \log y, t\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf 69.9%

      \[\leadsto \color{blue}{x \cdot \log y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+63} \lor \neg \left(t \leq 66000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

3. Taylor expanded in x around inf 95.0%

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

4. Final simplification 95.0%

   \[\leadsto x \cdot \log y - t \]
5. Add Preprocessing

Alternative 5: 48.8% accurate, 105.5× 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 95.7%

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

3. Taylor expanded in t around inf 51.4%

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

   1. mul-1-neg 51.4%

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

5. Simplified 51.4%

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

6. Final simplification 51.4%

   \[\leadsto -t \]
7. Add Preprocessing

Alternative 6: 3.3% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := -1.0

Derivation

1. Initial program 95.7%

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

   1. sub-neg 95.7%

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

   2. add-sqr-sqrt 52.4%

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

   3. sqrt-unprod 57.7%

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

   4. sqr-neg 57.7%

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

   5. sqrt-unprod 20.3%

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

   6. add-sqr-sqrt 44.4%

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

   7. expm1-log1p-u 27.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + t\right)\right)} \]

   8. add-sqr-sqrt 10.7%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \color{blue}{\sqrt{t} \cdot \sqrt{t}}\right)\right) \]

   9. sqrt-unprod 40.8%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \color{blue}{\sqrt{t \cdot t}}\right)\right) \]

   10. sqr-neg 40.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \sqrt{\color{blue}{\left(-t\right) \cdot \left(-t\right)}}\right)\right) \]

   11. sqrt-unprod 43.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \color{blue}{\sqrt{-t} \cdot \sqrt{-t}}\right)\right) \]

   12. add-sqr-sqrt 57.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + \color{blue}{\left(-t\right)}\right)\right) \]

   13. sub-neg 57.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t}\right)\right) \]

   14. +-commutative 57.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(z \cdot \log \left(1 - y\right) + x \cdot \log y\right)} - t\right)\right) \]

   15. associate-+r- 57.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{z \cdot \log \left(1 - y\right) + \left(x \cdot \log y - t\right)}\right)\right) \]

   16. sub-neg 57.3%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(z \cdot \log \color{blue}{\left(1 + \left(-y\right)\right)} + \left(x \cdot \log y - t\right)\right)\right) \]

   17. log1p-udef 48.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} + \left(x \cdot \log y - t\right)\right)\right) \]

   18. fma-udef 48.8%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right)}\right)\right) \]

4. Applied egg-rr 22.7%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(z, \mathsf{log1p}\left(y\right), \mathsf{fma}\left(x, \log y, t\right)\right)\right)\right)} \]

5. Taylor expanded in x around inf 0.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \log y + -1 \cdot \log \left(\frac{1}{x}\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 0.0%

      \[\leadsto \mathsf{expm1}\left(\log \log y + \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)}\right) \]

   2. log-rec 0.0%

      \[\leadsto \mathsf{expm1}\left(\log \log y + \left(-\color{blue}{\left(-\log x\right)}\right)\right) \]

   3. remove-double-neg 0.0%

      \[\leadsto \mathsf{expm1}\left(\log \log y + \color{blue}{\log x}\right) \]

   4. log-prod 19.4%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(\log y \cdot x\right)}\right) \]

   5. *-commutative 19.4%

      \[\leadsto \mathsf{expm1}\left(\log \color{blue}{\left(x \cdot \log y\right)}\right) \]

7. Simplified 19.4%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(x \cdot \log y\right)}\right) \]
8. Step-by-step derivation

   1. expm1-udef 19.4%

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

   2. add-exp-log 36.8%

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

   3. flip3-- 13.8%

      \[\leadsto \color{blue}{\frac{{\left(x \cdot \log y\right)}^{3} - {1}^{3}}{\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right) + \left(1 \cdot 1 + \left(x \cdot \log y\right) \cdot 1\right)}} \]

   4. metadata-eval 13.8%

      \[\leadsto \frac{{\left(x \cdot \log y\right)}^{3} - \color{blue}{1}}{\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right) + \left(1 \cdot 1 + \left(x \cdot \log y\right) \cdot 1\right)} \]

   5. pow2 13.8%

      \[\leadsto \frac{{\left(x \cdot \log y\right)}^{3} - 1}{\color{blue}{{\left(x \cdot \log y\right)}^{2}} + \left(1 \cdot 1 + \left(x \cdot \log y\right) \cdot 1\right)} \]

   6. metadata-eval 13.8%

      \[\leadsto \frac{{\left(x \cdot \log y\right)}^{3} - 1}{{\left(x \cdot \log y\right)}^{2} + \left(\color{blue}{1} + \left(x \cdot \log y\right) \cdot 1\right)} \]

   7. *-commutative 13.8%

      \[\leadsto \frac{{\left(x \cdot \log y\right)}^{3} - 1}{{\left(x \cdot \log y\right)}^{2} + \left(1 + \color{blue}{1 \cdot \left(x \cdot \log y\right)}\right)} \]

   8. *-un-lft-identity 13.8%

      \[\leadsto \frac{{\left(x \cdot \log y\right)}^{3} - 1}{{\left(x \cdot \log y\right)}^{2} + \left(1 + \color{blue}{x \cdot \log y}\right)} \]

9. Applied egg-rr 13.8%

   \[\leadsto \color{blue}{\frac{{\left(x \cdot \log y\right)}^{3} - 1}{{\left(x \cdot \log y\right)}^{2} + \left(1 + x \cdot \log y\right)}} \]

10. Taylor expanded in x around 0 3.1%

    \[\leadsto \color{blue}{-1} \]

11. Final simplification 3.1%

    \[\leadsto -1 \]
12. Add Preprocessing

Developer target: 87.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024031 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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