Logistic regression 2

Percentage Accurate: 99.3% → 99.2%

Time: 11.9s

Alternatives: 8

Speedup: 1.8×

• could not determine a ground truth

  1. x = 1.9918804948538048e+55
  2. y = -7.635535196540887e+195

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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (/ 1.0 (- (log1p (exp x)) (* x y)))))

C

double code(double x, double y) {
	return 1.0 / (1.0 / (log1p(exp(x)) - (x * y)));
}

Java

public static double code(double x, double y) {
	return 1.0 / (1.0 / (Math.log1p(Math.exp(x)) - (x * y)));
}

Python

def code(x, y):
	return 1.0 / (1.0 / (math.log1p(math.exp(x)) - (x * y)))

Julia

function code(x, y)
	return Float64(1.0 / Float64(1.0 / Float64(log1p(exp(x)) - Float64(x * y))))
end

Wolfram

code[x_, y_] := N[(1.0 / N[(1.0 / N[(N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-- N/A

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

   6. /-lowering-/.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y}} \]

   9. exp-lowering-exp.f64 N/A

      \[\leadsto \frac{1}{\frac{1}{\mathsf{log1p}\left(\color{blue}{e^{x}}\right) - x \cdot y}} \]

   10. *-lowering-*.f64 99.5

       \[\leadsto \frac{1}{\frac{1}{\mathsf{log1p}\left(e^{x}\right) - \color{blue}{x \cdot y}}} \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y}}} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := 0 - x \cdot y\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (- 0.0 (* x y))))
   (if (<= t_0 2e-5) t_1 (if (<= t_0 1.0) (log 2.0) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = 0.0 - (x * y);
	double tmp;
	if (t_0 <= 2e-5) {
		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((1.0d0 + exp(x))) - (x * y)
    t_1 = 0.0d0 - (x * y)
    if (t_0 <= 2d-5) 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((1.0 + Math.exp(x))) - (x * y);
	double t_1 = 0.0 - (x * y);
	double tmp;
	if (t_0 <= 2e-5) {
		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((1.0 + math.exp(x))) - (x * y)
	t_1 = 0.0 - (x * y)
	tmp = 0
	if t_0 <= 2e-5:
		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(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(0.0 - Float64(x * y))
	tmp = 0.0
	if (t_0 <= 2e-5)
		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((1.0 + exp(x))) - (x * y);
	t_1 = 0.0 - (x * y);
	tmp = 0.0;
	if (t_0 <= 2e-5)
		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[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-5], 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)) < 2.00000000000000016e-5 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(y\right), 0\right)} \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

      6. --lowering--.f64 96.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

   5. Simplified 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

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

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 96.3

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

   7. Applied egg-rr 96.3%

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

   if 2.00000000000000016e-5 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. log-lowering-log.f64 98.3

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

   5. Simplified 98.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 1:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;0 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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]

Derivation

1. Initial program 99.4%

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

Alternative 4: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -710000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -710000.0)
   (- 0.0 (* x y))
   (fma x (- (fma x 0.125 0.5) y) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -710000.0) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = fma(x, (fma(x, 0.125, 0.5) - y), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -710000.0)
		tmp = Float64(0.0 - Float64(x * y));
	else
		tmp = fma(x, Float64(fma(x, 0.125, 0.5) - y), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -710000.0], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(x * 0.125 + 0.5), $MachinePrecision] - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1e5

   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(y\right), 0\right)} \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

      6. --lowering--.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

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

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -7.1e5 < x

   1. Initial program 99.1%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 98.5

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

   5. Simplified 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -710000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4) (- 0.0 (* x y)) (fma x (- 0.5 y) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = fma(x, (0.5 - y), log(2.0));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999

   1. Initial program 99.3%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(y\right), 0\right)} \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

      6. --lowering--.f64 98.8

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

   5. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

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

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 98.8

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

   7. Applied egg-rr 98.8%

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

   if -1.3999999999999999 < 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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 98.9

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

   5. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -710000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -710000.0) (- 0.0 (* x y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -710000.0) {
		tmp = 0.0 - (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 <= (-710000.0d0)) then
        tmp = 0.0d0 - (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 <= -710000.0) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -710000.0:
		tmp = 0.0 - (x * y)
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -710000.0)
		tmp = Float64(0.0 - 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 <= -710000.0)
		tmp = 0.0 - (x * y);
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -710000.0], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1e5

   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. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(y\right), 0\right)} \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

      6. --lowering--.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

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

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -7.1e5 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. log-lowering-log.f64 98.2

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

   5. Simplified 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -710000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% accurate, 23.6× speedup

Math

\[\begin{array}{l} \\ 0 - x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 0.0 (* x y)))

C

double code(double x, double y) {
	return 0.0 - (x * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0 - (x * y)
end function

Java

public static double code(double x, double y) {
	return 0.0 - (x * y);
}

Python

def code(x, y):
	return 0.0 - (x * y)

Julia

function code(x, y)
	return Float64(0.0 - Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0 - (x * y);
end

Wolfram

code[x_, y_] := N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. +-rgt-identity N/A

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(y\right), 0\right)} \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

   6. --lowering--.f64 52.9

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

5. Simplified 52.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

   2. sub0-neg N/A

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

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

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

   4. neg-lowering-neg.f64 N/A

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

   5. *-lowering-*.f64 52.9

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

7. Applied egg-rr 52.9%

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

8. Final simplification 52.9%

   \[\leadsto 0 - x \cdot y \]
9. Add Preprocessing

Alternative 8: 2.3% accurate, 35.3× speedup

Math

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

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(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 99.4%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. +-rgt-identity N/A

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

   4. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{neg}\left(y\right), 0\right)} \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

   6. --lowering--.f64 52.9

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0 - y}, 0\right) \]

5. Simplified 52.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

   2. sub0-neg N/A

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

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

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

   4. neg-lowering-neg.f64 N/A

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

   5. *-lowering-*.f64 52.9

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

7. Applied egg-rr 52.9%

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

9. Applied egg-rr 2.3%

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

10. Final simplification 2.3%

    \[\leadsto x \cdot y \]
11. 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 2024196 
(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)))