Logistic regression 2

Percentage Accurate: 99.2% → 99.2%

Time: 5.9s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.0760065876923373e+155
  2. y = -16351.486090193901

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -44:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -44.0) (* (- x) y) (fma (fma 0.125 x (- 0.5 y)) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -44.0) {
		tmp = -x * y;
	} else {
		tmp = fma(fma(0.125, x, (0.5 - y)), x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -44.0)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(fma(0.125, x, Float64(0.5 - y)), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -44.0], N[((-x) * y), $MachinePrecision], N[(N[(0.125 * x + N[(0.5 - y), $MachinePrecision]), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -44

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -44 < x

   1. Initial program 98.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) + \log 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot x} + \log 2 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y, x, \log 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{8} \cdot x + \frac{1}{2}\right)} - y, x, \log 2\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{8} \cdot x + \left(\frac{1}{2} - y\right)}, x, \log 2\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{8}, x, \frac{1}{2} - y\right)}, x, \log 2\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, x, \color{blue}{\frac{1}{2} - y}\right), x, \log 2\right) \]

      8. lower-log.f64 98.9

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \color{blue}{\log 2}\right) \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \log 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{if}\;t\_0 \leq 0.0004 \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))))
   (if (or (<= t_0 0.0004) (not (<= t_0 1.0)))
     (* (- x) y)
     (fma 0.5 x (log 2.0)))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double tmp;
	if ((t_0 <= 0.0004) || !(t_0 <= 1.0)) {
		tmp = -x * y;
	} else {
		tmp = fma(0.5, x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	tmp = 0.0
	if ((t_0 <= 0.0004) || !(t_0 <= 1.0))
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(0.5, x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0004], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[((-x) * y), $MachinePrecision], N[(0.5 * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 4.00000000000000019e-4 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 97.7%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 4.00000000000000019e-4 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 1

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 99.6

         \[\leadsto \mathsf{fma}\left(0.5 - y, x, \color{blue}{\log 2}\right) \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \log 2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(0.5, x, \log 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.0004 \lor \neg \left(\log \left(1 + e^{x}\right) - x \cdot y \leq 1\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{if}\;t\_0 \leq 0.0004 \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))))
   (if (or (<= t_0 0.0004) (not (<= t_0 1.0))) (* (- x) y) (log 2.0))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double tmp;
	if ((t_0 <= 0.0004) || !(t_0 <= 1.0)) {
		tmp = -x * y;
	} else {
		tmp = log(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((1.0d0 + exp(x))) - (x * y)
    if ((t_0 <= 0.0004d0) .or. (.not. (t_0 <= 1.0d0))) then
        tmp = -x * y
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (x * y);
	double tmp;
	if ((t_0 <= 0.0004) || !(t_0 <= 1.0)) {
		tmp = -x * y;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (x * y)
	tmp = 0
	if (t_0 <= 0.0004) or not (t_0 <= 1.0):
		tmp = -x * y
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	tmp = 0.0
	if ((t_0 <= 0.0004) || !(t_0 <= 1.0))
		tmp = Float64(Float64(-x) * y);
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = log((1.0 + exp(x))) - (x * y);
	tmp = 0.0;
	if ((t_0 <= 0.0004) || ~((t_0 <= 1.0)))
		tmp = -x * y;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0004], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[((-x) * y), $MachinePrecision], N[Log[2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 4.00000000000000019e-4 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 97.7%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if 4.00000000000000019e-4 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 1

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\log \left(1 + e^{x}\right)}^{2}}} + \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      7. pow2 N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right)}} + \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      8. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{\log \left(1 + e^{x}\right)} \cdot \sqrt{\log \left(1 + e^{x}\right)}} + \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log \left(1 + e^{x}\right)}, \sqrt{\log \left(1 + e^{x}\right)}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\log \left(1 + e^{x}\right)}}, \sqrt{\log \left(1 + e^{x}\right)}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right) \]

      11. lift-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\log \left(1 + e^{x}\right)}}, \sqrt{\log \left(1 + e^{x}\right)}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right) \]

      12. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\log \color{blue}{\left(1 + e^{x}\right)}}, \sqrt{\log \left(1 + e^{x}\right)}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right) \]

      13. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\mathsf{log1p}\left(e^{x}\right)}}, \sqrt{\log \left(1 + e^{x}\right)}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right) \]

      14. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(e^{x}\right)}, \color{blue}{\sqrt{\log \left(1 + e^{x}\right)}}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right) \]

      15. lift-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(e^{x}\right)}, \sqrt{\color{blue}{\log \left(1 + e^{x}\right)}}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(e^{x}\right)}, \sqrt{\log \color{blue}{\left(1 + e^{x}\right)}}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right) \]

      17. lower-log1p.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(e^{x}\right)}, \sqrt{\color{blue}{\mathsf{log1p}\left(e^{x}\right)}}, \left(\mathsf{neg}\left(x\right)\right) \cdot y\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(e^{x}\right)}, \sqrt{\mathsf{log1p}\left(e^{x}\right)}, \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot y}\right) \]

      19. lower-neg.f64 98.5

          \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(e^{x}\right)}, \sqrt{\mathsf{log1p}\left(e^{x}\right)}, \color{blue}{\left(-x\right)} \cdot y\right) \]

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(e^{x}\right)}, \sqrt{\mathsf{log1p}\left(e^{x}\right)}, \left(-x\right) \cdot y\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2} \]
   6. Step-by-step derivation

      1. lower-log.f64 99.0

         \[\leadsto \color{blue}{\log 2} \]

   7. Applied rewrites 99.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.0004 \lor \neg \left(\log \left(1 + e^{x}\right) - x \cdot y \leq 1\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 5: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - y, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4) (* (- x) y) (fma (- 0.5 y) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = -x * y;
	} else {
		tmp = fma((0.5 - y), x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(Float64(0.5 - y), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[((-x) * y), $MachinePrecision], N[(N[(0.5 - y), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.3999999999999999 < x

   1. Initial program 98.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 98.8

         \[\leadsto \mathsf{fma}\left(0.5 - y, x, \color{blue}{\log 2}\right) \]

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -44:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -44.0) (* (- x) y) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -44.0) {
		tmp = -x * y;
	} else {
		tmp = log(2.0) - (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-44.0d0)) then
        tmp = -x * y
    else
        tmp = log(2.0d0) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -44.0) {
		tmp = -x * y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -44.0:
		tmp = -x * y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -44.0)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = Float64(log(2.0) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -44.0)
		tmp = -x * y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -44.0], N[((-x) * y), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -44

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -44 < x

   1. Initial program 98.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \log \color{blue}{2} - x \cdot y \]
   4. Step-by-step derivation

   5. Applied rewrites 98.3%

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

Alternative 7: 51.9% accurate, 26.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 50.8

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

5. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\left(-x\right) \cdot y} \]
6. Add Preprocessing

Alternative 8: 3.5% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 0.5 x))

C

double code(double x, double y) {
	return 0.5 * x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 * x
end function

Java

public static double code(double x, double y) {
	return 0.5 * x;
}

Python

def code(x, y):
	return 0.5 * x

Julia

function code(x, y)
	return Float64(0.5 * x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 * x;
end

Wolfram

code[x_, y_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 98.9%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-log.f64 84.1

      \[\leadsto \mathsf{fma}\left(0.5 - y, x, \color{blue}{\log 2}\right) \]

5. Applied rewrites 84.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 36.1%

   \[\leadsto \left(0.5 - y\right) \cdot \color{blue}{x} \]

9. Taylor expanded in y around 0

   \[\leadsto \frac{1}{2} \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 4.0%

    \[\leadsto 0.5 \cdot x \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0)
   (- (log (+ 1.0 (exp x))) (* x y))
   (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = log((1.0 + exp(x))) - (x * y);
	} else {
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0d0) then
        tmp = log((1.0d0 + exp(x))) - (x * y)
    else
        tmp = log((1.0d0 + exp(-x))) - (-x * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = Math.log((1.0 + Math.exp(x))) - (x * y);
	} else {
		tmp = Math.log((1.0 + Math.exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0:
		tmp = math.log((1.0 + math.exp(x))) - (x * y)
	else:
		tmp = math.log((1.0 + math.exp(-x))) - (-x * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0)
		tmp = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y));
	else
		tmp = Float64(log(Float64(1.0 + exp(Float64(-x)))) - Float64(Float64(-x) * Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0)
		tmp = log((1.0 + exp(x))) - (x * y);
	else
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0], N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[((-x) * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024326 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y)))))

  (- (log (+ 1.0 (exp x))) (* x y)))