Logarithmic Transform

Percentage Accurate: 41.6% → 99.2%

Time: 23.0s

Alternatives: 12

Speedup: 19.8×

• Could not converge on a ground truth

  1. c = -8.857398474829321e+247
  2. x = 6.638694295586305e+193
  3. y = 479949007828.88696

Specification

Math

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

FPCore

(FPCore (c x y)
 :precision binary64
 (* c (log (+ 1.0 (* (- (pow E x) 1.0) y)))))

C

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

Java

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

Python

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

Julia

function code(c, x, y)
	return Float64(c * log(Float64(1.0 + Float64(Float64((exp(1) ^ x) - 1.0) * y))))
end

MATLAB

function tmp = code(c, x, y)
	tmp = c * log((1.0 + (((2.71828182845904523536 ^ x) - 1.0) * y)));
end

Wolfram

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

Initial Program: 41.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (c x y)
 :precision binary64
 (* c (log (+ 1.0 (* (- (pow E x) 1.0) y)))))

C

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

Java

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

Python

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

Julia

function code(c, x, y)
	return Float64(c * log(Float64(1.0 + Float64(Float64((exp(1) ^ x) - 1.0) * y))))
end

MATLAB

function tmp = code(c, x, y)
	tmp = c * log((1.0 + (((2.71828182845904523536 ^ x) - 1.0) * y)));
end

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{3} \cdot c, 0.3333333333333333, \left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{4} \cdot y\right) \cdot c\right) \cdot -0.25\right), y, \left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot c\right) \cdot -0.5\right), y, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -3.6e-33)
   (* (log1p (* (expm1 x) y)) c)
   (if (<= y 1.15)
     (*
      (fma
       (fma
        (fma
         (* (pow (expm1 x) 3.0) c)
         0.3333333333333333
         (* (* (* (pow (expm1 x) 4.0) y) c) -0.25))
        y
        (* (* (pow (expm1 x) 2.0) c) -0.5))
       y
       (* (expm1 x) c))
      y)
     (*
      (log1p
       (*
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)
        y))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -3.6e-33) {
		tmp = log1p((expm1(x) * y)) * c;
	} else if (y <= 1.15) {
		tmp = fma(fma(fma((pow(expm1(x), 3.0) * c), 0.3333333333333333, (((pow(expm1(x), 4.0) * y) * c) * -0.25)), y, ((pow(expm1(x), 2.0) * c) * -0.5)), y, (expm1(x) * c)) * y;
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -3.6e-33)
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	elseif (y <= 1.15)
		tmp = Float64(fma(fma(fma(Float64((expm1(x) ^ 3.0) * c), 0.3333333333333333, Float64(Float64(Float64((expm1(x) ^ 4.0) * y) * c) * -0.25)), y, Float64(Float64((expm1(x) ^ 2.0) * c) * -0.5)), y, Float64(expm1(x) * c)) * y);
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -3.6e-33], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.15], N[(N[(N[(N[(N[(N[Power[N[(Exp[x] - 1), $MachinePrecision], 3.0], $MachinePrecision] * c), $MachinePrecision] * 0.3333333333333333 + N[(N[(N[(N[Power[N[(Exp[x] - 1), $MachinePrecision], 4.0], $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * y + N[(N[(N[Power[N[(Exp[x] - 1), $MachinePrecision], 2.0], $MachinePrecision] * c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * y + N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.60000000000000034e-33

   1. Initial program 48.0%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]

   if -3.60000000000000034e-33 < y < 1.1499999999999999

   1. Initial program 44.0%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + y \cdot \left(\frac{-1}{4} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{4}\right)\right) + \frac{1}{3} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{3} \cdot c, 0.3333333333333333, \left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{4} \cdot y\right) \cdot c\right) \cdot -0.25\right), y, \left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot c\right) \cdot -0.5\right), y, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]

   if 1.1499999999999999 < y

   1. Initial program 6.2%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)} \cdot y\right) \cdot c \]

   7. Applied rewrites 97.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \cdot c \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{3} \cdot c, 0.3333333333333333, \left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{4} \cdot y\right) \cdot c\right) \cdot -0.25\right), y, \left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot c\right) \cdot -0.5\right), y, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-33} \lor \neg \left(y \leq 5.8 \cdot 10^{-105}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -5e-33) (not (<= y 5.8e-105)))
   (* (log1p (* (expm1 x) y)) c)
   (* (* c y) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -5e-33) || !(y <= 5.8e-105)) {
		tmp = log1p((expm1(x) * y)) * c;
	} else {
		tmp = (c * y) * expm1(x);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -5e-33) || !(y <= 5.8e-105)) {
		tmp = Math.log1p((Math.expm1(x) * y)) * c;
	} else {
		tmp = (c * y) * Math.expm1(x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -5e-33) or not (y <= 5.8e-105):
		tmp = math.log1p((math.expm1(x) * y)) * c
	else:
		tmp = (c * y) * math.expm1(x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -5e-33) || !(y <= 5.8e-105))
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	else
		tmp = Float64(Float64(c * y) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -5e-33], N[Not[LessEqual[y, 5.8e-105]], $MachinePrecision]], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000028e-33 or 5.80000000000000007e-105 < y

   1. Initial program 33.5%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]
   6. Step-by-step derivation

   7. Applied rewrites 98.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]

   if -5.00000000000000028e-33 < y < 5.80000000000000007e-105

   1. Initial program 46.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{\log \mathsf{E}\left(\right) \cdot x} - 1\right) \]

      5. log-E N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{1 \cdot x} - 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{x \cdot 1} - 1\right) \]

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 99.8

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-33} \lor \neg \left(y \leq 5.8 \cdot 10^{-105}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1560:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1560.0)
   (* (log1p (* x y)) c)
   (if (<= y 0.2)
     (* (* c y) (expm1 x))
     (*
      (log1p
       (*
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)
        y))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1560.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 0.2) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1560.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 0.2)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1560.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.2], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1560

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]
   6. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      3. lower-expm1.f64 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      4. *-rgt-identity 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      5. *-commutative 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      6. log-E 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      7. pow-to-exp 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

   7. Applied rewrites 63.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]

   if -1560 < y < 0.20000000000000001

   1. Initial program 42.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{\log \mathsf{E}\left(\right) \cdot x} - 1\right) \]

      5. log-E N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{1 \cdot x} - 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{x \cdot 1} - 1\right) \]

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 98.9

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]

   if 0.20000000000000001 < y

   1. Initial program 6.2%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)} \cdot y\right) \cdot c \]

   7. Applied rewrites 97.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \cdot c \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1560:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1560:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1560.0)
   (* (log1p (* x y)) c)
   (if (<= y 0.2)
     (* (* c y) (expm1 x))
     (* (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1560.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 0.2) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = log1p(((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1560.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 0.2)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1560.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.2], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1560

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]
   6. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      3. lower-expm1.f64 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      4. *-rgt-identity 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      5. *-commutative 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      6. log-E 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      7. pow-to-exp 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

   7. Applied rewrites 63.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]

   if -1560 < y < 0.20000000000000001

   1. Initial program 42.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{\log \mathsf{E}\left(\right) \cdot x} - 1\right) \]

      5. log-E N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{1 \cdot x} - 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{x \cdot 1} - 1\right) \]

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 98.9

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]

   if 0.20000000000000001 < y

   1. Initial program 6.2%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right)} \cdot y\right) \cdot c \]
   6. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\left(\color{blue}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      3. lower-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\left(\color{blue}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      5. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      6. log-E N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \cdot c \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} \cdot 1 + \frac{1}{6} \cdot x\right)\right) \cdot x\right) \cdot y\right) \cdot c \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} \cdot {1}^{2} + \frac{1}{6} \cdot x\right)\right) \cdot x\right) \cdot y\right) \cdot c \]

      11. log-E N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \frac{1}{6} \cdot x\right)\right) \cdot x\right) \cdot y\right) \cdot c \]

      12. *-rgt-identity N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \frac{1}{6} \cdot \left(x \cdot 1\right)\right)\right) \cdot x\right) \cdot y\right) \cdot c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \frac{1}{6} \cdot \left(x \cdot {1}^{3}\right)\right)\right) \cdot x\right) \cdot y\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right) \cdot x\right) \cdot y\right) \cdot c \]

      15. +-commutative N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot x\right) \cdot y\right) \cdot c \]

      16. log-E N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot x\right) \cdot y\right) \cdot c \]

   7. Applied rewrites 97.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \cdot c \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1560:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1560:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(0.5 \cdot x, y, y\right) \cdot x\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1560.0)
   (* (log1p (* x y)) c)
   (if (<= y 0.2)
     (* (* c y) (expm1 x))
     (* (log1p (* (fma (* 0.5 x) y y) x)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1560.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 0.2) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = log1p((fma((0.5 * x), y, y) * x)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1560.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 0.2)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(log1p(Float64(fma(Float64(0.5 * x), y, y) * x)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1560.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.2], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(0.5 * x), $MachinePrecision] * y + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1560

   1. Initial program 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]
   6. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      3. lower-expm1.f64 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      4. *-rgt-identity 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      5. *-commutative 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      6. log-E 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      7. pow-to-exp 63.5

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

   7. Applied rewrites 63.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]

   if -1560 < y < 0.20000000000000001

   1. Initial program 42.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{\log \mathsf{E}\left(\right) \cdot x} - 1\right) \]

      5. log-E N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{1 \cdot x} - 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{x \cdot 1} - 1\right) \]

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 98.9

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]

   if 0.20000000000000001 < y

   1. Initial program 6.2%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right)}\right) \cdot c \]
   6. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right)\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right)\right) \cdot c \]

      3. lower-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right)\right) \cdot c \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right)\right) \cdot c \]

      5. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right)\right) \cdot c \]

      6. log-E N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right)\right) \cdot c \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot \left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right)\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\left(y + \frac{1}{2} \cdot \left(x \cdot y\right)\right) \cdot \color{blue}{x}\right) \cdot c \]

      9. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(\left(y \cdot 1 + \frac{1}{2} \cdot \left(x \cdot y\right)\right) \cdot x\right) \cdot c \]

      10. log-E N/A

          \[\leadsto \mathsf{log1p}\left(\left(y \cdot \log \mathsf{E}\left(\right) + \frac{1}{2} \cdot \left(x \cdot y\right)\right) \cdot x\right) \cdot c \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{log1p}\left(\left(y \cdot \log \mathsf{E}\left(\right) + \left(\frac{1}{2} \cdot x\right) \cdot y\right) \cdot x\right) \cdot c \]

      12. *-rgt-identity N/A

          \[\leadsto \mathsf{log1p}\left(\left(y \cdot \log \mathsf{E}\left(\right) + \left(\frac{1}{2} \cdot x\right) \cdot \left(y \cdot 1\right)\right) \cdot x\right) \cdot c \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{log1p}\left(\left(y \cdot \log \mathsf{E}\left(\right) + \left(\frac{1}{2} \cdot x\right) \cdot \left(y \cdot {1}^{2}\right)\right) \cdot x\right) \cdot c \]

      14. log-E N/A

          \[\leadsto \mathsf{log1p}\left(\left(y \cdot \log \mathsf{E}\left(\right) + \left(\frac{1}{2} \cdot x\right) \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot x\right) \cdot c \]

      15. associate-*r* N/A

          \[\leadsto \mathsf{log1p}\left(\left(y \cdot \log \mathsf{E}\left(\right) + \frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right) \cdot x\right) \cdot c \]

      16. +-commutative N/A

          \[\leadsto \mathsf{log1p}\left(\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) + y \cdot \log \mathsf{E}\left(\right)\right) \cdot x\right) \cdot c \]

      17. lower-*.f64 N/A

          \[\leadsto \mathsf{log1p}\left(\left(\frac{1}{2} \cdot \left(x \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) + y \cdot \log \mathsf{E}\left(\right)\right) \cdot \color{blue}{x}\right) \cdot c \]

   7. Applied rewrites 97.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot x, y, y\right) \cdot x}\right) \cdot c \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1560:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(0.5 \cdot x, y, y\right) \cdot x\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-148} \lor \neg \left(x \leq 8.8 \cdot 10^{-167}\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -4.8e-7)
   (* (* (expm1 x) y) c)
   (if (or (<= x -5e-148) (not (<= x 8.8e-167)))
     (* (log1p (* x y)) c)
     (* (* c x) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -4.8e-7) {
		tmp = (expm1(x) * y) * c;
	} else if ((x <= -5e-148) || !(x <= 8.8e-167)) {
		tmp = log1p((x * y)) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -4.8e-7) {
		tmp = (Math.expm1(x) * y) * c;
	} else if ((x <= -5e-148) || !(x <= 8.8e-167)) {
		tmp = Math.log1p((x * y)) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -4.8e-7:
		tmp = (math.expm1(x) * y) * c
	elif (x <= -5e-148) or not (x <= 8.8e-167):
		tmp = math.log1p((x * y)) * c
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -4.8e-7)
		tmp = Float64(Float64(expm1(x) * y) * c);
	elseif ((x <= -5e-148) || !(x <= 8.8e-167))
		tmp = Float64(log1p(Float64(x * y)) * c);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -4.8e-7], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], If[Or[LessEqual[x, -5e-148], N[Not[LessEqual[x, 8.8e-167]], $MachinePrecision]], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999957e-7

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in y around 0

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

      1. lift-expm1.f64 N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      3. lower-expm1.f64 N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      4. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      5. *-commutative N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      6. log-E N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      7. pow-to-exp N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      8. lower-expm1.f64 N/A

         \[\leadsto \left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c \]

      9. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c \]

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      14. lift-*.f64 71.2

          \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \color{blue}{y}\right) \cdot c \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      16. *-rgt-identity 71.2

          \[\leadsto \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c \]

   7. Applied rewrites 71.2%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot y\right)} \cdot c \]

   if -4.79999999999999957e-7 < x < -4.9999999999999999e-148 or 8.7999999999999999e-167 < x

   1. Initial program 19.2%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]
   6. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      3. lower-expm1.f64 93.3

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      4. *-rgt-identity 93.3

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      5. *-commutative 93.3

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      6. log-E 93.3

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      7. pow-to-exp 93.3

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

   7. Applied rewrites 93.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]

   if -4.9999999999999999e-148 < x < 8.7999999999999999e-167

   1. Initial program 40.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot 1\right) \]

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 93.1

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

   5. Applied rewrites 93.1%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 93.1

         \[\leadsto \left(c \cdot x\right) \cdot y \]

   7. Applied rewrites 93.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-148} \lor \neg \left(x \leq 8.8 \cdot 10^{-167}\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+226}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-92}:\\ \;\;\;\;\left(y \cdot x\right) \cdot c\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(c \cdot y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* x y)))))
   (if (<= y -2.3e+226)
     t_0
     (if (<= y -5.8e-92)
       (* (* y x) c)
       (if (<= y 4.1e+105)
         (* x (fma c y (* x (* y (fma 0.5 c (* -0.5 (* c y)))))))
         t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((x * y));
	double tmp;
	if (y <= -2.3e+226) {
		tmp = t_0;
	} else if (y <= -5.8e-92) {
		tmp = (y * x) * c;
	} else if (y <= 4.1e+105) {
		tmp = x * fma(c, y, (x * (y * fma(0.5, c, (-0.5 * (c * y))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(x * y)))
	tmp = 0.0
	if (y <= -2.3e+226)
		tmp = t_0;
	elseif (y <= -5.8e-92)
		tmp = Float64(Float64(y * x) * c);
	elseif (y <= 4.1e+105)
		tmp = Float64(x * fma(c, y, Float64(x * Float64(y * fma(0.5, c, Float64(-0.5 * Float64(c * y)))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e+226], t$95$0, If[LessEqual[y, -5.8e-92], N[(N[(y * x), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 4.1e+105], N[(x * N[(c * y + N[(x * N[(y * N[(0.5 * c + N[(-0.5 * N[(c * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.29999999999999995e226 or 4.1000000000000002e105 < y

   1. Initial program 29.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

         \[\leadsto c \cdot \log \left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot \color{blue}{y}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \log \left(\left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot \color{blue}{y}\right) \]

      3. pow-to-exp N/A

         \[\leadsto c \cdot \log \left(\left(e^{\log \mathsf{E}\left(\right) \cdot x} - 1\right) \cdot y\right) \]

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto c \cdot \log \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \]

      7. lower-*.f64 82.0

         \[\leadsto c \cdot \log \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \]

   5. Applied rewrites 82.0%

      \[\leadsto c \cdot \log \color{blue}{\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \left(x \cdot y\right) \]
   7. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto c \cdot \log \left(x \cdot y\right) \]

      2. *-rgt-identity N/A

         \[\leadsto c \cdot \log \left(x \cdot y\right) \]

      3. lower-expm1.f64 59.3

         \[\leadsto c \cdot \log \left(x \cdot y\right) \]

      4. *-rgt-identity 59.3

         \[\leadsto c \cdot \log \left(x \cdot y\right) \]

      5. *-commutative 59.3

         \[\leadsto c \cdot \log \left(x \cdot y\right) \]

      6. log-E 59.3

         \[\leadsto c \cdot \log \left(x \cdot y\right) \]

      7. pow-to-exp 59.3

         \[\leadsto c \cdot \log \left(x \cdot y\right) \]

   8. Applied rewrites 59.3%

      \[\leadsto c \cdot \log \left(x \cdot y\right) \]

   if -2.29999999999999995e226 < y < -5.79999999999999969e-92

   1. Initial program 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

         \[\leadsto \left(x \cdot y\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \left(x \cdot y\right) \cdot c \]

      3. lower-expm1.f64 N/A

         \[\leadsto \left(x \cdot y\right) \cdot c \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot y\right) \cdot c \]

      5. *-commutative N/A

         \[\leadsto \left(x \cdot y\right) \cdot c \]

      6. log-E N/A

         \[\leadsto \left(x \cdot y\right) \cdot c \]

      7. pow-to-exp N/A

         \[\leadsto \left(x \cdot y\right) \cdot c \]

      8. *-commutative N/A

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

      9. lower-*.f64 55.2

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

   7. Applied rewrites 55.2%

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

   if -5.79999999999999969e-92 < y < 4.1000000000000002e105

   1. Initial program 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]
   6. Step-by-step derivation

   7. Applied rewrites 86.9%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + y \cdot \left(\frac{-1}{4} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{4}\right)\right) + \frac{1}{3} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   10. Applied rewrites 96.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{3} \cdot c, 0.3333333333333333, \left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{4} \cdot y\right) \cdot c\right) \cdot -0.25\right), y, \left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot c\right) \cdot -0.5\right), y, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(\frac{1}{2}, c, \frac{-1}{2} \cdot \left(c \cdot y\right)\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(\frac{1}{2}, c, \frac{-1}{2} \cdot \left(c \cdot y\right)\right)\right)\right) \]

       8. lower-*.f64 77.1

          \[\leadsto x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(c \cdot y\right)\right)\right)\right) \]

   13. Applied rewrites 77.1%

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

Alternative 8: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1560 \lor \neg \left(y \leq 0.2\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1560.0) (not (<= y 0.2)))
   (* (log1p (* x y)) c)
   (* (* c y) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1560.0) || !(y <= 0.2)) {
		tmp = log1p((x * y)) * c;
	} else {
		tmp = (c * y) * expm1(x);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1560.0) || !(y <= 0.2)) {
		tmp = Math.log1p((x * y)) * c;
	} else {
		tmp = (c * y) * Math.expm1(x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -1560.0) or not (y <= 0.2):
		tmp = math.log1p((x * y)) * c
	else:
		tmp = (c * y) * math.expm1(x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1560.0) || !(y <= 0.2))
		tmp = Float64(log1p(Float64(x * y)) * c);
	else
		tmp = Float64(Float64(c * y) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1560.0], N[Not[LessEqual[y, 0.2]], $MachinePrecision]], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1560 or 0.20000000000000001 < y

   1. Initial program 34.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]
   6. Step-by-step derivation

      1. lift-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      3. lower-expm1.f64 76.2

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      4. *-rgt-identity 76.2

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      5. *-commutative 76.2

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      6. log-E 76.2

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

      7. pow-to-exp 76.2

         \[\leadsto \mathsf{log1p}\left(x \cdot y\right) \cdot c \]

   7. Applied rewrites 76.2%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]

   if -1560 < y < 0.20000000000000001

   1. Initial program 42.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{\log \mathsf{E}\left(\right) \cdot x} - 1\right) \]

      5. log-E N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{1 \cdot x} - 1\right) \]

      6. *-commutative N/A

         \[\leadsto \left(c \cdot y\right) \cdot \left(e^{x \cdot 1} - 1\right) \]

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 98.9

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1560 \lor \neg \left(y \leq 0.2\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -1e-24) (* (* (expm1 x) y) c) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -1e-24) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -1e-24) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -1e-24:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -1e-24)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -1e-24], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999924e-25

   1. Initial program 57.0%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in y around 0

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

      1. lift-expm1.f64 N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      3. lower-expm1.f64 N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      4. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      5. *-commutative N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      6. log-E N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      7. pow-to-exp N/A

         \[\leadsto \left(y \cdot \left(e^{x} - 1\right)\right) \cdot c \]

      8. lower-expm1.f64 N/A

         \[\leadsto \left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c \]

      9. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c \]

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      14. lift-*.f64 70.6

          \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \color{blue}{y}\right) \cdot c \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      16. *-rgt-identity 70.6

          \[\leadsto \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c \]

   7. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\left(\mathsf{expm1}\left(x\right) \cdot y\right)} \cdot c \]

   if -9.99999999999999924e-25 < x

   1. Initial program 29.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot 1\right) \]

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 81.7

         \[\leadsto \left(c \cdot x\right) \cdot y \]

   7. Applied rewrites 81.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(c, y, x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(c \cdot y\right)\right), x \cdot \left(y \cdot \mathsf{fma}\left(0.16666666666666666, c, y \cdot \mathsf{fma}\left(-0.5, c, 0.3333333333333333 \cdot \left(c \cdot y\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 2.6e-29)
   (*
    x
    (fma
     c
     y
     (*
      x
      (fma
       y
       (fma 0.5 c (* -0.5 (* c y)))
       (*
        x
        (*
         y
         (fma
          0.16666666666666666
          c
          (* y (fma -0.5 c (* 0.3333333333333333 (* c y)))))))))))
   (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.6e-29) {
		tmp = x * fma(c, y, (x * fma(y, fma(0.5, c, (-0.5 * (c * y))), (x * (y * fma(0.16666666666666666, c, (y * fma(-0.5, c, (0.3333333333333333 * (c * y))))))))));
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 2.6e-29)
		tmp = Float64(x * fma(c, y, Float64(x * fma(y, fma(0.5, c, Float64(-0.5 * Float64(c * y))), Float64(x * Float64(y * fma(0.16666666666666666, c, Float64(y * fma(-0.5, c, Float64(0.3333333333333333 * Float64(c * y)))))))))));
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 2.6e-29], N[(x * N[(c * y + N[(x * N[(y * N[(0.5 * c + N[(-0.5 * N[(c * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * N[(0.16666666666666666 * c + N[(y * N[(-0.5 * c + N[(0.3333333333333333 * N[(c * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.6000000000000002e-29

   1. Initial program 49.8%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]
   6. Step-by-step derivation

   7. Applied rewrites 95.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + y \cdot \left(\frac{-1}{4} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{4}\right)\right) + \frac{1}{3} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   10. Applied rewrites 76.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{3} \cdot c, 0.3333333333333333, \left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{4} \cdot y\right) \cdot c\right) \cdot -0.25\right), y, \left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot c\right) \cdot -0.5\right), y, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]

   11. Taylor expanded in x around 0

       \[\leadsto x \cdot \color{blue}{\left(c \cdot y + x \cdot \left(x \cdot \left(y \cdot \left(\frac{1}{6} \cdot c + y \cdot \left(\frac{-1}{2} \cdot c + \frac{1}{3} \cdot \left(c \cdot y\right)\right)\right)\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \frac{1}{2} \cdot c\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto x \cdot \left(c \cdot y + \color{blue}{x \cdot \left(x \cdot \left(y \cdot \left(\frac{1}{6} \cdot c + y \cdot \left(\frac{-1}{2} \cdot c + \frac{1}{3} \cdot \left(c \cdot y\right)\right)\right)\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \frac{1}{2} \cdot c\right)\right)}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(c, y, x \cdot \left(x \cdot \left(y \cdot \left(\frac{1}{6} \cdot c + y \cdot \left(\frac{-1}{2} \cdot c + \frac{1}{3} \cdot \left(c \cdot y\right)\right)\right)\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \frac{1}{2} \cdot c\right)\right)\right) \]

       3. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(c, y, x \cdot \left(x \cdot \left(y \cdot \left(\frac{1}{6} \cdot c + y \cdot \left(\frac{-1}{2} \cdot c + \frac{1}{3} \cdot \left(c \cdot y\right)\right)\right)\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \frac{1}{2} \cdot c\right)\right)\right) \]

   13. Applied rewrites 61.8%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(c, y, x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(c \cdot y\right)\right), x \cdot \left(y \cdot \mathsf{fma}\left(0.16666666666666666, c, y \cdot \mathsf{fma}\left(-0.5, c, 0.3333333333333333 \cdot \left(c \cdot y\right)\right)\right)\right)\right)\right)} \]

   if 2.6000000000000002e-29 < c

   1. Initial program 12.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot 1\right) \]

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 59.1

         \[\leadsto \left(c \cdot x\right) \cdot y \]

   7. Applied rewrites 59.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(c, y, x \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(c \cdot y\right)\right), x \cdot \left(y \cdot \mathsf{fma}\left(0.16666666666666666, c, y \cdot \mathsf{fma}\left(-0.5, c, 0.3333333333333333 \cdot \left(c \cdot y\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(c \cdot y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 2.6e-29)
   (* x (fma c y (* x (* y (fma 0.5 c (* -0.5 (* c y)))))))
   (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.6e-29) {
		tmp = x * fma(c, y, (x * (y * fma(0.5, c, (-0.5 * (c * y))))));
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 2.6e-29)
		tmp = Float64(x * fma(c, y, Float64(x * Float64(y * fma(0.5, c, Float64(-0.5 * Float64(c * y)))))));
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 2.6e-29], N[(x * N[(c * y + N[(x * N[(y * N[(0.5 * c + N[(-0.5 * N[(c * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.6000000000000002e-29

   1. Initial program 49.8%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]
   6. Step-by-step derivation

   7. Applied rewrites 95.2%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{x}\right) \cdot y\right) \cdot c \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + y \cdot \left(\frac{-1}{4} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{4}\right)\right) + \frac{1}{3} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   10. Applied rewrites 76.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{3} \cdot c, 0.3333333333333333, \left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{4} \cdot y\right) \cdot c\right) \cdot -0.25\right), y, \left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot c\right) \cdot -0.5\right), y, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(\frac{1}{2}, c, \frac{-1}{2} \cdot \left(c \cdot y\right)\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(\frac{1}{2}, c, \frac{-1}{2} \cdot \left(c \cdot y\right)\right)\right)\right) \]

       8. lower-*.f64 63.0

          \[\leadsto x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(c \cdot y\right)\right)\right)\right) \]

   13. Applied rewrites 63.0%

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

   if 2.6000000000000002e-29 < c

   1. Initial program 12.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot 1\right) \]

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 59.1

         \[\leadsto \left(c \cdot x\right) \cdot y \]

   7. Applied rewrites 59.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(c, y, x \cdot \left(y \cdot \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(c \cdot y\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.6% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (c * x) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.5%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. log-E N/A

      \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot 1\right) \]

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 57.0

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

5. Applied rewrites 57.0%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 57.0

      \[\leadsto \left(c \cdot x\right) \cdot y \]

7. Applied rewrites 57.0%

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

8. Final simplification 57.0%

   \[\leadsto \left(c \cdot x\right) \cdot y \]
9. Add Preprocessing

Developer Target 1: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \end{array} \]

FPCore

(FPCore (c x y) :precision binary64 (* c (log1p (* (expm1 x) y))))

C

double code(double c, double x, double y) {
	return c * log1p((expm1(x) * y));
}

Java

public static double code(double c, double x, double y) {
	return c * Math.log1p((Math.expm1(x) * y));
}

Python

def code(c, x, y):
	return c * math.log1p((math.expm1(x) * y))

Julia

function code(c, x, y)
	return Float64(c * log1p(Float64(expm1(x) * y)))
end

Wolfram

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

Reproduce

herbie shell --seed 2025061 
(FPCore (c x y)
  :name "Logarithmic Transform"
  :precision binary64

  :alt
  (* c (log1p (* (expm1 x) y)))

  (* c (log (+ 1.0 (* (- (pow E x) 1.0) y)))))