Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 9.1s

Alternatives: 12

Speedup: 1.7×

• could not determine a ground truth

  1. x = 2.094383640746923e+247
  2. y = -1.646004974134286e+286

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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), x \cdot x, \mathsf{fma}\left(x, 0.5 - y, \log 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6)
   (* x (- y))
   (fma
    (fma -0.005208333333333333 (* x x) 0.125)
    (* x x)
    (fma x (- 0.5 y) (log 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6) {
		tmp = x * -y;
	} else {
		tmp = fma(fma(-0.005208333333333333, (x * x), 0.125), (x * x), fma(x, (0.5 - y), log(2.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6)
		tmp = Float64(x * Float64(-y));
	else
		tmp = fma(fma(-0.005208333333333333, Float64(x * x), 0.125), Float64(x * x), fma(x, Float64(0.5 - y), log(2.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6], N[(x * (-y)), $MachinePrecision], N[(N[(-0.005208333333333333 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * N[(0.5 - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009

   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. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -2.60000000000000009 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right) + \log 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y, \log 2\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) + \frac{1}{2}\right)} - y, \log 2\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) + \left(\frac{1}{2} - y\right)}, \log 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}, \frac{1}{2} - y\right)}, \log 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{192} \cdot {x}^{2} + \frac{1}{8}}, \frac{1}{2} - y\right), \log 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{-1}{192}, {x}^{2}, \frac{1}{8}\right)}, \frac{1}{2} - y\right), \log 2\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{192}, \color{blue}{x \cdot x}, \frac{1}{8}\right), \frac{1}{2} - y\right), \log 2\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{192}, \color{blue}{x \cdot x}, \frac{1}{8}\right), \frac{1}{2} - y\right), \log 2\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right), \color{blue}{\frac{1}{2} - y}\right), \log 2\right) \]

      11. lower-log.f64 99.1

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), 0.5 - y\right), \color{blue}{\log 2}\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), 0.5 - y\right), \log 2\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \left(\frac{-1}{192} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{8}\right) + \left(\frac{1}{2} - y\right)\right) + \log 2 \]

      2. lift-fma.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right)} + \left(\frac{1}{2} - y\right)\right) + \log 2 \]

      3. lift--.f64 N/A

         \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right) + \color{blue}{\left(\frac{1}{2} - y\right)}\right) + \log 2 \]

      4. lift-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right), \frac{1}{2} - y\right)} + \log 2 \]

      5. lift-log.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right), \frac{1}{2} - y\right) + \color{blue}{\log 2} \]

      6. lift-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right) + \left(\frac{1}{2} - y\right)\right)} + \log 2 \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right)\right) \cdot x + \left(\frac{1}{2} - y\right) \cdot x\right)} + \log 2 \]

      8. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right)\right) \cdot x + \left(\left(\frac{1}{2} - y\right) \cdot x + \log 2\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(x \cdot \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right)\right) \cdot x + \left(\color{blue}{x \cdot \left(\frac{1}{2} - y\right)} + \log 2\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right) \cdot x\right)} \cdot x + \left(x \cdot \left(\frac{1}{2} - y\right) + \log 2\right) \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right) \cdot \left(x \cdot x\right)} + \left(x \cdot \left(\frac{1}{2} - y\right) + \log 2\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right) \cdot \color{blue}{\left(x \cdot x\right)} + \left(x \cdot \left(\frac{1}{2} - y\right) + \log 2\right) \]

   7. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), x \cdot x, \mathsf{fma}\left(x, 0.5 - y, \log 2\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := x \cdot \left(-y\right)\\ \mathbf{if}\;t\_0 \leq 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (* x (- y))))
   (if (<= t_0 1e-15) t_1 (if (<= t_0 500.0) (fma x 0.5 (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 = x * -y;
	double tmp;
	if (t_0 <= 1e-15) {
		tmp = t_1;
	} else if (t_0 <= 500.0) {
		tmp = fma(x, 0.5, 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(x * Float64(-y))
	tmp = 0.0
	if (t_0 <= 1e-15)
		tmp = t_1;
	elseif (t_0 <= 500.0)
		tmp = fma(x, 0.5, log(2.0));
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(x * (-y)), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-15], t$95$1, If[LessEqual[t$95$0, 500.0], N[(x * 0.5 + 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)) < 1.0000000000000001e-15 or 500 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 98.5%

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

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 99.5

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

   5. Simplified 99.5%

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

   if 1.0000000000000001e-15 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 500

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log \left(1 + e^{x}\right)} \]
   4. Step-by-step derivation

      1. lower-log1p.f64 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

      2. lower-exp.f64 97.7

         \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{x}}\right) \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + \frac{1}{2} \cdot x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \log 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + \log 2 \]

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 95.6

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

   8. Simplified 95.6%

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

Alternative 3: 97.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := x \cdot \left(-y\right)\\ \mathbf{if}\;t\_0 \leq 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (* x (- y))))
   (if (<= t_0 1e-15) t_1 (if (<= t_0 500.0) (log (+ x 2.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = x * -y;
	double tmp;
	if (t_0 <= 1e-15) {
		tmp = t_1;
	} else if (t_0 <= 500.0) {
		tmp = log((x + 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 = x * -y
    if (t_0 <= 1d-15) then
        tmp = t_1
    else if (t_0 <= 500.0d0) then
        tmp = log((x + 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 = x * -y;
	double tmp;
	if (t_0 <= 1e-15) {
		tmp = t_1;
	} else if (t_0 <= 500.0) {
		tmp = Math.log((x + 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 = x * -y
	tmp = 0
	if t_0 <= 1e-15:
		tmp = t_1
	elif t_0 <= 500.0:
		tmp = math.log((x + 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(x * Float64(-y))
	tmp = 0.0
	if (t_0 <= 1e-15)
		tmp = t_1;
	elseif (t_0 <= 500.0)
		tmp = log(Float64(x + 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 = x * -y;
	tmp = 0.0;
	if (t_0 <= 1e-15)
		tmp = t_1;
	elseif (t_0 <= 500.0)
		tmp = log((x + 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[(x * (-y)), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-15], t$95$1, If[LessEqual[t$95$0, 500.0], N[Log[N[(x + 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)) < 1.0000000000000001e-15 or 500 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 98.5%

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

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 99.5

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

   5. Simplified 99.5%

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

   if 1.0000000000000001e-15 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 500

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log \left(1 + e^{x}\right)} \]
   4. Step-by-step derivation

      1. lower-log1p.f64 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

      2. lower-exp.f64 97.7

         \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{x}}\right) \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 95.5

         \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x}\right) \]

   8. Simplified 95.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x}\right) \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \log \left(1 + \color{blue}{\left(1 + x\right)}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto \color{blue}{\log \left(1 + \left(1 + x\right)\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \log \left(1 + \color{blue}{\left(1 + x\right)}\right) \]

      4. associate-+r+ N/A

         \[\leadsto \log \color{blue}{\left(\left(1 + 1\right) + x\right)} \]

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 95.5

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

   10. Applied egg-rr 95.5%

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

Alternative 4: 97.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := x \cdot \left(-y\right)\\ \mathbf{if}\;t\_0 \leq 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (* x (- y))))
   (if (<= t_0 1e-15) t_1 (if (<= t_0 500.0) (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 = x * -y;
	double tmp;
	if (t_0 <= 1e-15) {
		tmp = t_1;
	} else if (t_0 <= 500.0) {
		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 = x * -y
    if (t_0 <= 1d-15) then
        tmp = t_1
    else if (t_0 <= 500.0d0) 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 = x * -y;
	double tmp;
	if (t_0 <= 1e-15) {
		tmp = t_1;
	} else if (t_0 <= 500.0) {
		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 = x * -y
	tmp = 0
	if t_0 <= 1e-15:
		tmp = t_1
	elif t_0 <= 500.0:
		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(x * Float64(-y))
	tmp = 0.0
	if (t_0 <= 1e-15)
		tmp = t_1;
	elseif (t_0 <= 500.0)
		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 = x * -y;
	tmp = 0.0;
	if (t_0 <= 1e-15)
		tmp = t_1;
	elseif (t_0 <= 500.0)
		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[(x * (-y)), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-15], t$95$1, If[LessEqual[t$95$0, 500.0], 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)) < 1.0000000000000001e-15 or 500 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 98.5%

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

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 99.5

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

   5. Simplified 99.5%

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

   if 1.0000000000000001e-15 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 500

   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 94.9

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

   5. Simplified 94.9%

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

Alternative 5: 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 99.2%

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

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (fma (- y) x (log1p (exp x))))

C

double code(double x, double y) {
	return fma(-y, x, log1p(exp(x)));
}

Julia

function code(x, y)
	return fma(Float64(-y), x, log1p(exp(x)))
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 99.2

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

   12. lift-log.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. lower-log1p.f64 99.2

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

4. Applied egg-rr 99.2%

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

Alternative 7: 99.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), 0.5 - y\right), \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6)
   (* x (- y))
   (fma
    x
    (fma x (fma -0.005208333333333333 (* x x) 0.125) (- 0.5 y))
    (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6) {
		tmp = x * -y;
	} else {
		tmp = fma(x, fma(x, fma(-0.005208333333333333, (x * x), 0.125), (0.5 - y)), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6)
		tmp = Float64(x * Float64(-y));
	else
		tmp = fma(x, fma(x, fma(-0.005208333333333333, Float64(x * x), 0.125), Float64(0.5 - y)), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6], N[(x * (-y)), $MachinePrecision], N[(x * N[(x * N[(-0.005208333333333333 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] + N[(0.5 - y), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009

   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. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -2.60000000000000009 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right) + \log 2} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y, \log 2\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) + \frac{1}{2}\right)} - y, \log 2\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) + \left(\frac{1}{2} - y\right)}, \log 2\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}, \frac{1}{2} - y\right)}, \log 2\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{-1}{192} \cdot {x}^{2} + \frac{1}{8}}, \frac{1}{2} - y\right), \log 2\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{-1}{192}, {x}^{2}, \frac{1}{8}\right)}, \frac{1}{2} - y\right), \log 2\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{192}, \color{blue}{x \cdot x}, \frac{1}{8}\right), \frac{1}{2} - y\right), \log 2\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{192}, \color{blue}{x \cdot x}, \frac{1}{8}\right), \frac{1}{2} - y\right), \log 2\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{192}, x \cdot x, \frac{1}{8}\right), \color{blue}{\frac{1}{2} - y}\right), \log 2\right) \]

      11. lower-log.f64 99.1

          \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), 0.5 - y\right), \color{blue}{\log 2}\right) \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), 0.5 - y\right), \log 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -15:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -15.0) (* 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 <= -15.0) {
		tmp = 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 <= -15.0)
		tmp = Float64(x * Float64(-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, -15.0], N[(x * (-y)), $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 < -15

   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. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -15 < x

   1. Initial program 98.7%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. lower-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. lower--.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. lower-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. lower-log.f64 98.8

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

   5. Simplified 98.8%

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

Alternative 9: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = 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(x * Float64(-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[(x * (-y)), $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 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. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -1.3999999999999999 < x

   1. Initial program 98.7%

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

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 98.2

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

   5. Simplified 98.2%

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

Alternative 10: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -170:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -170.0) (* x (- y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -170.0) {
		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 <= (-170.0d0)) 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 <= -170.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -170.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -170.0)
		tmp = Float64(x * Float64(-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 <= -170.0)
		tmp = x * -y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -170.0], 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 < -170

   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. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Simplified 100.0%

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

   if -170 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-log.f64 97.7

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

   5. Simplified 97.7%

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

Alternative 11: 50.5% accurate, 26.5× speedup

Math

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

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 * Float64(-y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 53.7

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

5. Simplified 53.7%

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

Alternative 12: 3.5% accurate, 35.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\log \left(1 + e^{x}\right)} \]
4. Step-by-step derivation

   1. lower-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

   2. lower-exp.f64 48.1

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{x}}\right) \]

5. Simplified 48.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\log 2 + \frac{1}{2} \cdot x} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \log 2} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + \log 2 \]

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 47.3

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

8. Simplified 47.3%

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

9. Taylor expanded in x around inf

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot \frac{1}{2}} \]

    2. lower-*.f64 3.4

       \[\leadsto \color{blue}{x \cdot 0.5} \]

11. Simplified 3.4%

    \[\leadsto \color{blue}{x \cdot 0.5} \]
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 2024207 
(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)))