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

Percentage Accurate: 89.2% → 99.8%

Time: 15.9s

Alternatives: 23

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: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 75.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\\ \mathbf{if}\;t\_2 \leq -8 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* (log (- 1.0 y)) (- z 1.0)) (* (log y) (- x 1.0)))))
   (if (<= t_2 -8e+51) t_1 (if (<= t_2 2e+19) (- (- y (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0));
	double tmp;
	if (t_2 <= -8e+51) {
		tmp = t_1;
	} else if (t_2 <= 2e+19) {
		tmp = (y - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = (log((1.0d0 - y)) * (z - 1.0d0)) + (log(y) * (x - 1.0d0))
    if (t_2 <= (-8d+51)) then
        tmp = t_1
    else if (t_2 <= 2d+19) then
        tmp = (y - log(y)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double t_2 = (Math.log((1.0 - y)) * (z - 1.0)) + (Math.log(y) * (x - 1.0));
	double tmp;
	if (t_2 <= -8e+51) {
		tmp = t_1;
	} else if (t_2 <= 2e+19) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = (math.log((1.0 - y)) * (z - 1.0)) + (math.log(y) * (x - 1.0))
	tmp = 0
	if t_2 <= -8e+51:
		tmp = t_1
	elif t_2 <= 2e+19:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(log(y) * Float64(x - 1.0)))
	tmp = 0.0
	if (t_2 <= -8e+51)
		tmp = t_1;
	elseif (t_2 <= 2e+19)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = (log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0));
	tmp = 0.0;
	if (t_2 <= -8e+51)
		tmp = t_1;
	elseif (t_2 <= 2e+19)
		tmp = (y - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -8e+51], t$95$1, If[LessEqual[t$95$2, 2e+19], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -8e51 or 2e19 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -8e51 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 2e19

   1. Initial program 89.3%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(y + -1 \cdot \log y\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 84.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right) \leq -8 \cdot 10^{+51}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right) \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right)\\ \mathbf{if}\;t\_2 \leq -8 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x))
        (t_2 (+ (* (log (- 1.0 y)) (- z 1.0)) (* (log y) (- x 1.0)))))
   (if (<= t_2 -8e+51) t_1 (if (<= t_2 2e+19) (- (- (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = (log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0));
	double tmp;
	if (t_2 <= -8e+51) {
		tmp = t_1;
	} else if (t_2 <= 2e+19) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = (log((1.0d0 - y)) * (z - 1.0d0)) + (log(y) * (x - 1.0d0))
    if (t_2 <= (-8d+51)) then
        tmp = t_1
    else if (t_2 <= 2d+19) then
        tmp = -log(y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double t_2 = (Math.log((1.0 - y)) * (z - 1.0)) + (Math.log(y) * (x - 1.0));
	double tmp;
	if (t_2 <= -8e+51) {
		tmp = t_1;
	} else if (t_2 <= 2e+19) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = (math.log((1.0 - y)) * (z - 1.0)) + (math.log(y) * (x - 1.0))
	tmp = 0
	if t_2 <= -8e+51:
		tmp = t_1
	elif t_2 <= 2e+19:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(log(Float64(1.0 - y)) * Float64(z - 1.0)) + Float64(log(y) * Float64(x - 1.0)))
	tmp = 0.0
	if (t_2 <= -8e+51)
		tmp = t_1;
	elseif (t_2 <= 2e+19)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = (log((1.0 - y)) * (z - 1.0)) + (log(y) * (x - 1.0));
	tmp = 0.0;
	if (t_2 <= -8e+51)
		tmp = t_1;
	elseif (t_2 <= 2e+19)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -8e+51], t$95$1, If[LessEqual[t$95$2, 2e+19], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -8e51 or 2e19 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -8e51 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 2e19

   1. Initial program 89.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right) \leq -8 \cdot 10^{+51}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot \left(z - 1\right) + \log y \cdot \left(x - 1\right) \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   3. sub-neg N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot y\right)\right) - t \]

   4. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot y\right)\right) - t \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y + \color{blue}{-1}\right) \cdot y\right)\right) - t \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}, y, -1\right)} \cdot y\right)\right) - t \]

   7. sub-neg N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, y, -1\right) \cdot y\right)\right) - t \]

   8. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), y, -1\right) \cdot y\right)\right) - t \]

   9. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y + \color{blue}{\frac{-1}{2}}, y, -1\right) \cdot y\right)\right) - t \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{4} \cdot y - \frac{1}{3}, y, \frac{-1}{2}\right)}, y, -1\right) \cdot y\right)\right) - t \]

   11. sub-neg N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right)\right) - t \]

   12. metadata-eval N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{4} \cdot y + \color{blue}{\frac{-1}{3}}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right)\right) - t \]

   13. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot \left(z - 1\right), y, 1 - z\right), y, -t\right)\right) \]
9. Add Preprocessing

Alternative 6: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot \left(z - 1\right), y, 1 - z\right), y, \log y \cdot \left(x - 1\right)\right) - t \]
7. Add Preprocessing

Alternative 7: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right) \cdot y}, z - 1, -t\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right) \cdot y}, z - 1, -t\right)\right) \]

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3} \cdot y - \frac{1}{2}, y, -1\right)} \cdot y, z - 1, -t\right)\right) \]

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 8: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(x - 1\right) - t\\ \mathbf{if}\;x - 1 \leq -1.00000000005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -0.999999998:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) (- x 1.0)) t)))
   (if (<= (- x 1.0) -1.00000000005)
     t_1
     (if (<= (- x 1.0) -0.999999998)
       (- (- (fma (- z 1.0) y (log y))) t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * (x - 1.0)) - t;
	double tmp;
	if ((x - 1.0) <= -1.00000000005) {
		tmp = t_1;
	} else if ((x - 1.0) <= -0.999999998) {
		tmp = -fma((z - 1.0), y, log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * Float64(x - 1.0)) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1.00000000005)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -0.999999998)
		tmp = Float64(Float64(-fma(Float64(z - 1.0), y, log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.00000000005], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -0.999999998], N[((-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.00000000005 or -0.999999997999999946 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 95.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if -1.00000000005 < (-.f64 x #s(literal 1 binary64)) < -0.999999997999999946

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. sub-neg N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1.00000000005:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \mathbf{elif}\;x - 1 \leq -0.999999998:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(x - 1, \log y, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot \left(z - 1\right), y, -t\right)\right) \]
9. Add Preprocessing

Alternative 10: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\left(z - 1\right) \cdot y, \mathsf{fma}\left(-0.5, y, -1\right), \log y \cdot \left(x - 1\right)\right) - t \]
7. Add Preprocessing

Alternative 11: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(x - 1\right)\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{elif}\;t \leq 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) (- x 1.0))))
   (if (<= t -1.75e-5)
     (- (fma (- x 1.0) (log y) y) t)
     (if (<= t 1e-37) (fma (- 1.0 z) y t_1) (- t_1 t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * (x - 1.0);
	double tmp;
	if (t <= -1.75e-5) {
		tmp = fma((x - 1.0), log(y), y) - t;
	} else if (t <= 1e-37) {
		tmp = fma((1.0 - z), y, t_1);
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * Float64(x - 1.0))
	tmp = 0.0
	if (t <= -1.75e-5)
		tmp = Float64(fma(Float64(x - 1.0), log(y), y) - t);
	elseif (t <= 1e-37)
		tmp = fma(Float64(1.0 - z), y, t_1);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.75e-5], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 1e-37], N[(N[(1.0 - z), $MachinePrecision] * y + t$95$1), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.7499999999999998e-5

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 96.8%

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

   if -1.7499999999999998e-5 < t < 1.00000000000000007e-37

   1. Initial program 86.3%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(1 - z, y, \log y \cdot \left(x - 1\right)\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

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

   if 1.00000000000000007e-37 < t

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 95.6

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

   5. Applied rewrites 95.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{elif}\;t \leq 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot \left(x - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 1e+222)
   (- (fma (- x 1.0) (log y) y) t)
   (fma (- 1.0 z) y (* (log y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 1e+222) {
		tmp = fma((x - 1.0), log(y), y) - t;
	} else {
		tmp = fma((1.0 - z), y, (log(y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 1e+222)
		tmp = Float64(fma(Float64(x - 1.0), log(y), y) - t);
	else
		tmp = fma(Float64(1.0 - z), y, Float64(log(y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+222], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(1.0 - z), $MachinePrecision] * y + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 1e222

   1. Initial program 94.3%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 93.2%

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

   if 1e222 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 54.1%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(1 - z, y, x \cdot \log y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 82.5%

      \[\leadsto \mathsf{fma}\left(1 - z, y, x \cdot \log y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, \log y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= x -1.0) t_1 (if (<= x 1.8e-15) (- (- y (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if (x <= -1.0) {
		tmp = t_1;
	} else if (x <= 1.8e-15) {
		tmp = (y - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (log(y) * x) - t
    if (x <= (-1.0d0)) then
        tmp = t_1
    else if (x <= 1.8d-15) then
        tmp = (y - log(y)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.log(y) * x) - t;
	double tmp;
	if (x <= -1.0) {
		tmp = t_1;
	} else if (x <= 1.8e-15) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - t
	tmp = 0
	if x <= -1.0:
		tmp = t_1
	elif x <= 1.8e-15:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_1;
	elseif (x <= 1.8e-15)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * x) - t;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_1;
	elseif (x <= 1.8e-15)
		tmp = (y - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$1, If[LessEqual[x, 1.8e-15], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.8000000000000001e-15 < x

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if -1 < x < 1.8000000000000001e-15

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 87.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(y + -1 \cdot \log y\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 86.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-15}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-neg.f64 99.1

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

7. Applied rewrites 99.1%

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

Alternative 15: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 N/A

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

   18. lower-neg.f64 99.1

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

5. Applied rewrites 99.1%

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

Alternative 16: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 N/A

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

   18. lower-neg.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 98.7%

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

Alternative 17: 64.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -9.8e-11) t_1 (if (<= x 7.8e+133) (fma (- 1.0 z) y (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -9.8e-11) {
		tmp = t_1;
	} else if (x <= 7.8e+133) {
		tmp = fma((1.0 - z), y, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -9.8e-11)
		tmp = t_1;
	elseif (x <= 7.8e+133)
		tmp = fma(Float64(1.0 - z), y, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -9.8e-11], t$95$1, If[LessEqual[x, 7.8e+133], N[(N[(1.0 - z), $MachinePrecision] * y + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7999999999999998e-11 or 7.80000000000000028e133 < x

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if -9.7999999999999998e-11 < x < 7.80000000000000028e133

   1. Initial program 88.3%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 64.6%

      \[\leadsto \mathsf{fma}\left(1 - z, y, -t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-11}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 88.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 N/A

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

   18. lower-neg.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 90.6%

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

Alternative 19: 88.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-log.f64 90.2

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

5. Applied rewrites 90.2%

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

6. Final simplification 90.2%

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

Alternative 20: 42.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-51}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left(1 - z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.3e-51) (- t) (if (<= t 1.75e-6) (* (- 1.0 z) y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.3e-51) {
		tmp = -t;
	} else if (t <= 1.75e-6) {
		tmp = (1.0 - z) * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.3d-51)) then
        tmp = -t
    else if (t <= 1.75d-6) then
        tmp = (1.0d0 - z) * y
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.3e-51) {
		tmp = -t;
	} else if (t <= 1.75e-6) {
		tmp = (1.0 - z) * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.3e-51:
		tmp = -t
	elif t <= 1.75e-6:
		tmp = (1.0 - z) * y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.3e-51)
		tmp = Float64(-t);
	elseif (t <= 1.75e-6)
		tmp = Float64(Float64(1.0 - z) * y);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.3e-51)
		tmp = -t;
	elseif (t <= 1.75e-6)
		tmp = (1.0 - z) * y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.3e-51], (-t), If[LessEqual[t, 1.75e-6], N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -2.30000000000000002e-51 or 1.74999999999999997e-6 < t

   1. Initial program 96.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -2.30000000000000002e-51 < t < 1.74999999999999997e-6

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 15.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 16.6%

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

Alternative 21: 42.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-51}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.3e-51) (- t) (if (<= t 1.75e-6) (* (- z) y) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.3e-51) {
		tmp = -t;
	} else if (t <= 1.75e-6) {
		tmp = -z * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.3d-51)) then
        tmp = -t
    else if (t <= 1.75d-6) then
        tmp = -z * y
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.3e-51) {
		tmp = -t;
	} else if (t <= 1.75e-6) {
		tmp = -z * y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.3e-51:
		tmp = -t
	elif t <= 1.75e-6:
		tmp = -z * y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.3e-51)
		tmp = Float64(-t);
	elseif (t <= 1.75e-6)
		tmp = Float64(Float64(-z) * y);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.3e-51)
		tmp = -t;
	elseif (t <= 1.75e-6)
		tmp = -z * y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.3e-51], (-t), If[LessEqual[t, 1.75e-6], N[((-z) * y), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -2.30000000000000002e-51 or 1.74999999999999997e-6 < t

   1. Initial program 96.0%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -2.30000000000000002e-51 < t < 1.74999999999999997e-6

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-neg.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 15.8%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-51}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 46.4% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 N/A

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

   18. lower-neg.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 45.6%

   \[\leadsto \mathsf{fma}\left(1 - z, y, -t\right) \]
9. Add Preprocessing

Alternative 23: 35.9% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 37.7

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

5. Applied rewrites 37.7%

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

Reproduce

herbie shell --seed 2024268 
(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))