Logarithmic Transform

Percentage Accurate: 42.1% → 98.7%

Time: 7.1s

Alternatives: 11

Speedup: 19.8×

• could not determine a ground truth

  1. c = -1.1709923941228969e-258
  2. x = 8.212226143267728e+25
  3. y = 3.920664719698745e-206

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: 42.1% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -3.05e-15)
   (* (log1p (* (expm1 x) y)) c)
   (if (<= y 1.55e-17)
     (* (* 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 <= -3.05e-15) {
		tmp = log1p((expm1(x) * y)) * c;
	} else if (y <= 1.55e-17) {
		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 <= -3.05e-15)
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	elseif (y <= 1.55e-17)
		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, -3.05e-15], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.55e-17], 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 < -3.04999999999999986e-15

   1. Initial program 45.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.7%

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

   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 99.7%

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

   if -3.04999999999999986e-15 < y < 1.5499999999999999e-17

   1. Initial program 43.0%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 1.5499999999999999e-17 < y

   1. Initial program 20.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 97.7%

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

   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 98.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -115.0)
   (* (log1p (* x y)) c)
   (if (<= y 1.55e-17)
     (* (* 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 <= -115.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 1.55e-17) {
		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 <= -115.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 1.55e-17)
		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, -115.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.55e-17], 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 < -115

   1. Initial program 45.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.8%

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

   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 66.0

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

      4. *-rgt-identity 66.0

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

      5. *-commutative 66.0

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

      6. log-E 66.0

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

      7. pow-to-exp 66.0

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

   7. Applied rewrites 66.0%

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

   if -115 < y < 1.5499999999999999e-17

   1. Initial program 43.0%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.5499999999999999e-17 < y

   1. Initial program 20.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 97.7%

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

   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 98.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -115.0)
   (* (log1p (* x y)) c)
   (if (<= y 1.55e-17)
     (* (* 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 <= -115.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 1.55e-17) {
		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 <= -115.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 1.55e-17)
		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, -115.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.55e-17], 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 < -115

   1. Initial program 45.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.8%

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

   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 66.0

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

      4. *-rgt-identity 66.0

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

      5. *-commutative 66.0

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

      6. log-E 66.0

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

      7. pow-to-exp 66.0

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

   7. Applied rewrites 66.0%

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

   if -115 < y < 1.5499999999999999e-17

   1. Initial program 43.0%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.5499999999999999e-17 < y

   1. Initial program 20.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 97.7%

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

   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 98.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -115.0)
   (* (log1p (* x y)) c)
   (if (<= y 1.55e-17)
     (* (* c y) (expm1 x))
     (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -115.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 1.55e-17) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = log1p(((fma(0.5, x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -115.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 1.55e-17)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(log1p(Float64(Float64(fma(0.5, x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -115.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.55e-17], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -115

   1. Initial program 45.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.8%

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

   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 66.0

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

      4. *-rgt-identity 66.0

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

      5. *-commutative 66.0

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

      6. log-E 66.0

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

      7. pow-to-exp 66.0

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

   7. Applied rewrites 66.0%

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

   if -115 < y < 1.5499999999999999e-17

   1. Initial program 43.0%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 1.5499999999999999e-17 < y

   1. Initial program 20.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. log-E N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 47.5

         \[\leadsto c \cdot \log \left(1 + \left(\mathsf{fma}\left(0.5 \cdot x, 1, 1\right) \cdot x\right) \cdot y\right) \]

   5. Applied rewrites 47.5%

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(\mathsf{fma}\left(0.5 \cdot x, 1, 1\right) \cdot x\right)} \cdot y\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.5

         \[\leadsto \color{blue}{\log \left(1 + \left(\mathsf{fma}\left(0.5 \cdot x, 1, 1\right) \cdot x\right) \cdot y\right) \cdot c} \]

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 98.0

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5 \cdot x, 1, 1\right) \cdot x\right) \cdot y\right)} \cdot c \]

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lower-fma.f64 98.0

          \[\leadsto \mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c \]

   7. Applied rewrites 98.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -15000000000:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-203} \lor \neg \left(x \leq 1.35 \cdot 10^{-241}\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -15000000000.0)
   (* (* (expm1 x) y) c)
   (if (or (<= x -5e-203) (not (<= x 1.35e-241)))
     (* (log1p (* x y)) c)
     (* (* c x) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -15000000000.0) {
		tmp = (expm1(x) * y) * c;
	} else if ((x <= -5e-203) || !(x <= 1.35e-241)) {
		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 <= -15000000000.0) {
		tmp = (Math.expm1(x) * y) * c;
	} else if ((x <= -5e-203) || !(x <= 1.35e-241)) {
		tmp = Math.log1p((x * y)) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -15000000000.0:
		tmp = (math.expm1(x) * y) * c
	elif (x <= -5e-203) or not (x <= 1.35e-241):
		tmp = math.log1p((x * y)) * c
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -15000000000.0)
		tmp = Float64(Float64(expm1(x) * y) * c);
	elseif ((x <= -5e-203) || !(x <= 1.35e-241))
		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, -15000000000.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], If[Or[LessEqual[x, -5e-203], N[Not[LessEqual[x, 1.35e-241]], $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 < -1.5e10

   1. Initial program 43.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.9%

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

   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 77.0

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 77.0

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

   7. Applied rewrites 77.0%

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

   if -1.5e10 < x < -5.0000000000000002e-203 or 1.35e-241 < x

   1. Initial program 37.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 94.6%

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

   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 93.1

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

      4. *-rgt-identity 93.1

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

      5. *-commutative 93.1

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

      6. log-E 93.1

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

      7. pow-to-exp 93.1

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

   7. Applied rewrites 93.1%

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

   if -5.0000000000000002e-203 < x < 1.35e-241

   1. Initial program 36.7%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15000000000:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-203} \lor \neg \left(x \leq 1.35 \cdot 10^{-241}\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115 \lor \neg \left(y \leq 1.55 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -115.0) (not (<= y 1.55e-17)))
   (* (log1p (* x y)) c)
   (* (* c y) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -115.0) || !(y <= 1.55e-17)) {
		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 <= -115.0) || !(y <= 1.55e-17)) {
		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 <= -115.0) or not (y <= 1.55e-17):
		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 <= -115.0) || !(y <= 1.55e-17))
		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, -115.0], N[Not[LessEqual[y, 1.55e-17]], $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 < -115 or 1.5499999999999999e-17 < y

   1. Initial program 33.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.8%

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

   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 80.8

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

      4. *-rgt-identity 80.8

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

      5. *-commutative 80.8

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

      6. log-E 80.8

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

      7. pow-to-exp 80.8

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

   7. Applied rewrites 80.8%

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

   if -115 < y < 1.5499999999999999e-17

   1. Initial program 43.0%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115 \lor \neg \left(y \leq 1.55 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-84}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(c \cdot x, 0.041666666666666664, 0.16666666666666666 \cdot c\right), x, 0.5 \cdot c\right), x, c\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -2e-84)
   (* (* (expm1 x) y) c)
   (*
    (*
     (fma
      (fma
       (fma (* c x) 0.041666666666666664 (* 0.16666666666666666 c))
       x
       (* 0.5 c))
      x
      c)
     x)
    y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -2e-84) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (fma(fma(fma((c * x), 0.041666666666666664, (0.16666666666666666 * c)), x, (0.5 * c)), x, c) * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -2e-84)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(fma(fma(fma(Float64(c * x), 0.041666666666666664, Float64(0.16666666666666666 * c)), x, Float64(0.5 * c)), x, c) * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -2e-84], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(N[(N[(N[(c * x), $MachinePrecision] * 0.041666666666666664 + N[(0.16666666666666666 * c), $MachinePrecision]), $MachinePrecision] * x + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * x + c), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-84

   1. Initial program 43.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.9%

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

   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 74.6

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 74.6

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

   7. Applied rewrites 74.6%

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

   if -2.0000000000000001e-84 < x

   1. Initial program 36.7%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 87.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. log-E N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 80.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \left(c + x \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \left(\frac{1}{2} \cdot c + x \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \left(\frac{1}{6} \cdot c + x \cdot \left(\frac{-7}{24} \cdot \left(c \cdot y\right) + \frac{1}{24} \cdot c\right)\right)\right)\right)\right)\right)\right) \cdot y \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(c + x \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \left(\frac{1}{2} \cdot c + x \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \left(\frac{1}{6} \cdot c + x \cdot \left(\frac{-7}{24} \cdot \left(c \cdot y\right) + \frac{1}{24} \cdot c\right)\right)\right)\right)\right)\right) \cdot x\right) \cdot y \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(c + x \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \left(\frac{1}{2} \cdot c + x \cdot \left(\frac{-1}{2} \cdot \left(c \cdot y\right) + \left(\frac{1}{6} \cdot c + x \cdot \left(\frac{-7}{24} \cdot \left(c \cdot y\right) + \frac{1}{24} \cdot c\right)\right)\right)\right)\right)\right) \cdot x\right) \cdot y \]

   10. Applied rewrites 76.3%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(c \cdot y, -0.5, \mathsf{fma}\left(\mathsf{fma}\left(c \cdot y, -0.5, \mathsf{fma}\left(\mathsf{fma}\left(-0.2916666666666667, c \cdot y, 0.041666666666666664 \cdot c\right), x, 0.16666666666666666 \cdot c\right)\right), x, 0.5 \cdot c\right)\right), x, c\right) \cdot x\right) \cdot y \]

   11. Taylor expanded in y around 0

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2} \cdot c + x \cdot \left(\frac{1}{24} \cdot \left(c \cdot x\right) + \frac{1}{6} \cdot c\right), x, c\right) \cdot x\right) \cdot y \]
   12. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(c \cdot x, \frac{1}{24}, \frac{1}{6} \cdot c\right), x, \frac{1}{2} \cdot c\right), x, c\right) \cdot x\right) \cdot y \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(c \cdot x, \frac{1}{24}, \frac{1}{6} \cdot c\right), x, \frac{1}{2} \cdot c\right), x, c\right) \cdot x\right) \cdot y \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(c \cdot x, \frac{1}{24}, \frac{1}{6} \cdot c\right), x, \frac{1}{2} \cdot c\right), x, c\right) \cdot x\right) \cdot y \]

       8. lift-*.f64 81.0

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(c \cdot x, 0.041666666666666664, 0.16666666666666666 \cdot c\right), x, 0.5 \cdot c\right), x, c\right) \cdot x\right) \cdot y \]

   13. Applied rewrites 81.0%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(c \cdot x, 0.041666666666666664, 0.16666666666666666 \cdot c\right), x, 0.5 \cdot c\right), x, c\right) \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c, \left(c \cdot y\right) \cdot -0.5\right) \cdot y, x, \mathsf{fma}\left(c \cdot y, -0.5, 0.5 \cdot c\right) \cdot y\right), x, c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.15e-82)
   (*
    (fma
     (fma
      (* (fma 0.16666666666666666 c (* (* c y) -0.5)) y)
      x
      (* (fma (* c y) -0.5 (* 0.5 c)) y))
     x
     (* c y))
    x)
   (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.15e-82) {
		tmp = fma(fma((fma(0.16666666666666666, c, ((c * y) * -0.5)) * y), x, (fma((c * y), -0.5, (0.5 * c)) * y)), x, (c * y)) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.15e-82)
		tmp = Float64(fma(fma(Float64(fma(0.16666666666666666, c, Float64(Float64(c * y) * -0.5)) * y), x, Float64(fma(Float64(c * y), -0.5, Float64(0.5 * c)) * y)), x, Float64(c * y)) * x);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.15e-82], N[(N[(N[(N[(N[(0.16666666666666666 * c + N[(N[(c * y), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * x + N[(N[(N[(c * y), $MachinePrecision] * -0.5 + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * x + N[(c * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.14999999999999998e-82

   1. Initial program 49.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 94.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. log-E N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 60.7%

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

   if 1.14999999999999998e-82 < c

   1. Initial program 17.8%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 53.4

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

   5. Applied rewrites 53.4%

      \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot 1\right)} \]
   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 53.4

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

   7. Applied rewrites 53.4%

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

Alternative 9: 60.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(c \cdot y, -0.5, 0.5 \cdot c\right) \cdot y, x, c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.15e-82)
   (* (fma (* (fma (* c y) -0.5 (* 0.5 c)) y) x (* c y)) x)
   (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.15e-82) {
		tmp = fma((fma((c * y), -0.5, (0.5 * c)) * y), x, (c * y)) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.15e-82)
		tmp = Float64(fma(Float64(fma(Float64(c * y), -0.5, Float64(0.5 * c)) * y), x, Float64(c * y)) * x);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.15e-82], N[(N[(N[(N[(N[(c * y), $MachinePrecision] * -0.5 + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * x + N[(c * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.14999999999999998e-82

   1. Initial program 49.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 94.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. log-E N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 60.5

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

   10. Applied rewrites 60.5%

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

   if 1.14999999999999998e-82 < c

   1. Initial program 17.8%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 53.4

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

   5. Applied rewrites 53.4%

      \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot 1\right)} \]
   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 53.4

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

   7. Applied rewrites 53.4%

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

Alternative 10: 58.8% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 2.3 \cdot 10^{+23}:\\ \;\;\;\;\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 2.3e+23) (* (* y x) c) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.3e+23) {
		tmp = (y * x) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (c <= 2.3d+23) then
        tmp = (y * x) * c
    else
        tmp = (c * x) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.3e+23) {
		tmp = (y * x) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 2.3e+23:
		tmp = (y * x) * c
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 2.3e+23)
		tmp = Float64(Float64(y * x) * c);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 2.3e+23)
		tmp = (y * x) * c;
	else
		tmp = (c * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 2.3e+23], N[(N[(y * x), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.3e23

   1. Initial program 46.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 95.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot c \]
   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 59.0

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

   7. Applied rewrites 59.0%

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

   if 2.3e23 < c

   1. Initial program 12.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 50.6

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

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot 1\right)} \]
   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 50.6

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

   7. Applied rewrites 50.6%

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

Alternative 11: 59.0% 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 39.2%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
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 56.8

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

5. Applied rewrites 56.8%

   \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot 1\right)} \]
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 56.8

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

7. Applied rewrites 56.8%

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

Developer Target 1: 93.8% 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 2025082 
(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)))))