Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 6.4s

Alternatives: 9

Speedup: 1.7×

• could not determine a ground truth

  1. x = 3.062427553075145e+186
  2. y = -9.760363440790999e-194

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: 99.3% 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: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -10.0) (* (- y) x) (fma (fma 0.125 x (- 0.5 y)) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -10.0) {
		tmp = -y * x;
	} 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 <= -10.0)
		tmp = Float64(Float64(-y) * x);
	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, -10.0], N[((-y) * x), $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 < -10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -10 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 2: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(e^{x} + 1\right) - y \cdot x\\ t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ (exp x) 1.0)) (* y x))) (t_1 (* (- y) x)))
   (if (<= t_0 1e-21) t_1 (if (<= t_0 1.0) (fma 0.5 x (log 2.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((exp(x) + 1.0)) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 1e-21) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma(0.5, x, log(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(log(Float64(exp(x) + 1.0)) - Float64(y * x))
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (t_0 <= 1e-21)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(0.5, x, log(2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-21], t$95$1, If[LessEqual[t$95$0, 1.0], N[(0.5 * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 9.99999999999999908e-22 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 9.99999999999999908e-22 < (-.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 99.8

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

   5. Applied rewrites 99.8%

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

   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 97.7%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{x} + 1\right) - y \cdot x \leq 10^{-21}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;\log \left(e^{x} + 1\right) - y \cdot x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(e^{x} + 1\right) - y \cdot x\\ t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq 10^{-21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ (exp x) 1.0)) (* y x))) (t_1 (* (- y) x)))
   (if (<= t_0 1e-21) t_1 (if (<= t_0 1.0) (log 2.0) t_1))))

C

double code(double x, double y) {
	double t_0 = log((exp(x) + 1.0)) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 1e-21) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = log(2.0);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = log((exp(x) + 1.0d0)) - (y * x)
    t_1 = -y * x
    if (t_0 <= 1d-21) then
        tmp = t_1
    else if (t_0 <= 1.0d0) then
        tmp = log(2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.log((Math.exp(x) + 1.0)) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 1e-21) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = Math.log(2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((math.exp(x) + 1.0)) - (y * x)
	t_1 = -y * x
	tmp = 0
	if t_0 <= 1e-21:
		tmp = t_1
	elif t_0 <= 1.0:
		tmp = math.log(2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(Float64(exp(x) + 1.0)) - Float64(y * x))
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (t_0 <= 1e-21)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = log(2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = log((exp(x) + 1.0)) - (y * x);
	t_1 = -y * x;
	tmp = 0.0;
	if (t_0 <= 1e-21)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = log(2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-21], t$95$1, If[LessEqual[t$95$0, 1.0], N[Log[2.0], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 9.99999999999999908e-22 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 99.3%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 9.99999999999999908e-22 < (-.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

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

      1. lower-log.f64 97.0

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

   5. Applied rewrites 97.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{x} + 1\right) - y \cdot x \leq 10^{-21}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;\log \left(e^{x} + 1\right) - y \cdot x \leq 1:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 5: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - y, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.36) (* (- y) x) (fma (- 0.5 y) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.36) {
		tmp = -y * x;
	} else {
		tmp = fma((0.5 - y), x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.36)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(Float64(0.5 - y), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.36], N[((-y) * x), $MachinePrecision], N[(N[(0.5 - y), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3600000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.3600000000000001 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 6: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -60:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -60.0) (* (- y) x) (fma (- y) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -60.0) {
		tmp = -y * x;
	} else {
		tmp = fma(-y, x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -60.0)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = fma(Float64(-y), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -60.0], N[((-y) * x), $MachinePrecision], N[((-y) * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -60

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -60 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 98.9%

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

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\log 2 - x \cdot y} \]

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 98.9

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

   7. Applied rewrites 98.9%

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

Alternative 7: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -60:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -60.0) (* (- y) x) (- (log 2.0) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -60.0) {
		tmp = -y * x;
	} else {
		tmp = log(2.0) - (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-60.0d0)) then
        tmp = -y * x
    else
        tmp = log(2.0d0) - (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -60.0) {
		tmp = -y * x;
	} else {
		tmp = Math.log(2.0) - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -60.0:
		tmp = -y * x
	else:
		tmp = math.log(2.0) - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -60.0)
		tmp = Float64(Float64(-y) * x);
	else
		tmp = Float64(log(2.0) - Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -60.0)
		tmp = -y * x;
	else
		tmp = log(2.0) - (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -60.0], N[((-y) * x), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -60

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -60 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -60:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.7% accurate, 26.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 54.5

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

5. Applied rewrites 54.5%

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

Alternative 9: 3.4% 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 99.6%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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 86.2

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

5. Applied rewrites 86.2%

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

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 41.3%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 3.4%

    \[\leadsto 0.5 \cdot x \]
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 2024263 
(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)))