Logarithmic Transform

Percentage Accurate: 41.7% → 99.0%

Time: 31.6s

Alternatives: 10

Speedup: 19.8×

• Could not converge on a ground truth

  1. c = -9.247615167061116e+223
  2. x = 4.779584817180552e+255
  3. y = 2.3494113490791507e+252

Specification

Math

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

FPCore

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

Initial Program: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-106} \lor \neg \left(y \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, {\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot y, \mathsf{expm1}\left(x\right)\right) \cdot \left(c \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -2e-106) (not (<= y 2e-11)))
   (* (log1p (* (expm1 x) y)) c)
   (* (fma -0.5 (* (pow (expm1 x) 2.0) y) (expm1 x)) (* c y))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -2e-106) || !(y <= 2e-11)) {
		tmp = log1p((expm1(x) * y)) * c;
	} else {
		tmp = fma(-0.5, (pow(expm1(x), 2.0) * y), expm1(x)) * (c * y);
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -2e-106) || !(y <= 2e-11))
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	else
		tmp = Float64(fma(-0.5, Float64((expm1(x) ^ 2.0) * y), expm1(x)) * Float64(c * y));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -2e-106], N[Not[LessEqual[y, 2e-11]], $MachinePrecision]], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(-0.5 * N[(N[Power[N[(Exp[x] - 1), $MachinePrecision], 2.0], $MachinePrecision] * y), $MachinePrecision] + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * N[(c * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999988e-106 or 1.99999999999999988e-11 < y

   1. Initial program 35.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 35.2

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 42.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.6

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

   6. Applied rewrites 99.6%

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

   if -1.99999999999999988e-106 < y < 1.99999999999999988e-11

   1. Initial program 35.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot y, -0.5, {\mathsf{E}\left(\right)}^{x} - 1\right)\right) \cdot y} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(-0.5, {\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot y, \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\left(c \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-106} \lor \neg \left(y \leq 2 \cdot 10^{-11}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, {\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot y, \mathsf{expm1}\left(x\right)\right) \cdot \left(c \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.1 \cdot 10^{+177} \lor \neg \left(c \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \mathsf{fma}\left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot y, -0.5, \mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= c 1.1e+177) (not (<= c 4e+231)))
   (* (log1p (* (expm1 x) y)) c)
   (*
    (* c (fma (* (pow (- (pow (E) x) 1.0) 2.0) y) -0.5 (* (fma 0.5 x 1.0) x)))
    y)))

Derivation

1. Split input into 2 regimes

2. if c < 1.0999999999999999e177 or 4.0000000000000002e231 < c

   1. Initial program 37.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.1

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 59.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.5

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 95.3

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

   6. Applied rewrites 95.3%

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

   if 1.0999999999999999e177 < c < 4.0000000000000002e231

   1. Initial program 4.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 10.5%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot y, -0.5, {\mathsf{E}\left(\right)}^{x} - 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.2%

      \[\leadsto \left(c \cdot \mathsf{fma}\left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot y, -0.5, \mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.1 \cdot 10^{+177} \lor \neg \left(c \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \mathsf{fma}\left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot y, -0.5, \mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.1 \cdot 10^{+177} \lor \neg \left(c \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(-0.2916666666666667, x, -0.5\right), x, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= c 1.1e+177) (not (<= c 4e+231)))
   (* (log1p (* (expm1 x) y)) c)
   (*
    (*
     (fma
      (* (* x x) y)
      (fma (fma -0.2916666666666667 x -0.5) x -0.5)
      (*
       (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
       x))
     c)
    y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((c <= 1.1e+177) || !(c <= 4e+231)) {
		tmp = log1p((expm1(x) * y)) * c;
	} else {
		tmp = (fma(((x * x) * y), fma(fma(-0.2916666666666667, x, -0.5), x, -0.5), (fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x)) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((c <= 1.1e+177) || !(c <= 4e+231))
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	else
		tmp = Float64(Float64(fma(Float64(Float64(x * x) * y), fma(fma(-0.2916666666666667, x, -0.5), x, -0.5), Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x)) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[c, 1.1e+177], N[Not[LessEqual[c, 4e+231]], $MachinePrecision]], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] * N[(N[(-0.2916666666666667 * x + -0.5), $MachinePrecision] * x + -0.5), $MachinePrecision] + N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.0999999999999999e177 or 4.0000000000000002e231 < c

   1. Initial program 37.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.1

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 59.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 59.5

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 95.3

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

   4. Applied rewrites 95.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

   6. Applied rewrites 95.3%

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

   if 1.0999999999999999e177 < c < 4.0000000000000002e231

   1. Initial program 4.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 10.5%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot y, -0.5, {\mathsf{E}\left(\right)}^{x} - 1\right)\right) \cdot y} \]

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \color{blue}{\left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right) + x \cdot \left(c \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) + x \cdot \left(c \cdot \left(x \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \left(y \cdot \left(\frac{1}{4} \cdot {\log \mathsf{E}\left(\right)}^{4} + \frac{1}{3} \cdot {\log \mathsf{E}\left(\right)}^{4}\right)\right) + \frac{1}{24} \cdot {\log \mathsf{E}\left(\right)}^{4}\right)\right)\right) + c \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{6} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 81.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(c \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, 0.16666666666666666\right), y, \left(\mathsf{fma}\left(0.5833333333333334 \cdot y, -0.5, 0.041666666666666664\right) \cdot y\right) \cdot x\right), x, \left(\mathsf{fma}\left(-0.5, y, 0.5\right) \cdot y\right) \cdot c\right), x, c \cdot y\right) \cdot \color{blue}{x} \]

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 93.2%

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

   12. Taylor expanded in c around 0

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

   14. Applied rewrites 93.2%

       \[\leadsto \left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(-0.2916666666666667, x, -0.5\right), x, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot c\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.1 \cdot 10^{+177} \lor \neg \left(c \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, \mathsf{fma}\left(\mathsf{fma}\left(-0.2916666666666667, x, -0.5\right), x, -0.5\right), \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.066:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-191} \lor \neg \left(x \leq 4.2 \cdot 10^{-208}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right) \cdot x, x, x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -0.066)
   (* (* (expm1 x) y) c)
   (if (or (<= x -2.6e-191) (not (<= x 4.2e-208)))
     (*
      (log1p
       (*
        y
        (fma
         (* (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x)
         x
         x)))
      c)
     (* (* x c) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -0.066) {
		tmp = (expm1(x) * y) * c;
	} else if ((x <= -2.6e-191) || !(x <= 4.2e-208)) {
		tmp = log1p((y * fma((fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5) * x), x, x))) * c;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -0.066)
		tmp = Float64(Float64(expm1(x) * y) * c);
	elseif ((x <= -2.6e-191) || !(x <= 4.2e-208))
		tmp = Float64(log1p(Float64(y * fma(Float64(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5) * x), x, x))) * c);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -0.066], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], If[Or[LessEqual[x, -2.6e-191], N[Not[LessEqual[x, 4.2e-208]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.066000000000000003

   1. Initial program 43.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.6

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 99.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 71.6

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

   7. Applied rewrites 71.6%

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

   if -0.066000000000000003 < x < -2.59999999999999986e-191 or 4.20000000000000024e-208 < x

   1. Initial program 26.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.0

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 28.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 28.6

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.9

          \[\leadsto \mathsf{log1p}\left(y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right) \cdot x, x, x\right)\right) \cdot c \]

   7. Applied rewrites 89.9%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right) \cdot x, x, x\right)}\right) \cdot c \]

   if -2.59999999999999986e-191 < x < 4.20000000000000024e-208

   1. Initial program 38.7%

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

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-commutative N/A

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

      16. *-lft-identity N/A

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

      17. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 95.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.066:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-191} \lor \neg \left(x \leq 4.2 \cdot 10^{-208}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right) \cdot x, x, x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right)\\ \mathbf{if}\;x \leq -0.066:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(t\_0, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{fma}\left(t\_0 \cdot x, x, x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)))
   (if (<= x -0.066)
     (* (* (expm1 x) y) c)
     (if (<= x -2.6e-191)
       (* (log1p (* (* (fma t_0 x 1.0) x) y)) c)
       (if (<= x 4.2e-208)
         (* (* x c) y)
         (* (log1p (* y (fma (* t_0 x) x x))) c))))))

C

double code(double c, double x, double y) {
	double t_0 = fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5);
	double tmp;
	if (x <= -0.066) {
		tmp = (expm1(x) * y) * c;
	} else if (x <= -2.6e-191) {
		tmp = log1p(((fma(t_0, x, 1.0) * x) * y)) * c;
	} else if (x <= 4.2e-208) {
		tmp = (x * c) * y;
	} else {
		tmp = log1p((y * fma((t_0 * x), x, x))) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5)
	tmp = 0.0
	if (x <= -0.066)
		tmp = Float64(Float64(expm1(x) * y) * c);
	elseif (x <= -2.6e-191)
		tmp = Float64(log1p(Float64(Float64(fma(t_0, x, 1.0) * x) * y)) * c);
	elseif (x <= 4.2e-208)
		tmp = Float64(Float64(x * c) * y);
	else
		tmp = Float64(log1p(Float64(y * fma(Float64(t_0 * x), x, x))) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision]}, If[LessEqual[x, -0.066], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, -2.6e-191], N[(N[Log[1 + N[(N[(N[(t$95$0 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 4.2e-208], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(y * N[(N[(t$95$0 * x), $MachinePrecision] * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.066000000000000003

   1. Initial program 43.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.6

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 99.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 71.6

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

   7. Applied rewrites 71.6%

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

   if -0.066000000000000003 < x < -2.59999999999999986e-191

   1. Initial program 22.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 22.8

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 26.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 26.7

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

   6. Applied rewrites 95.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 94.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \cdot c \]

   if -2.59999999999999986e-191 < x < 4.20000000000000024e-208

   1. Initial program 38.7%

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

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-commutative N/A

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

      16. *-lft-identity N/A

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

      17. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 95.8%

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

   if 4.20000000000000024e-208 < x

   1. Initial program 28.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 28.6

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 30.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 30.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 86.3

          \[\leadsto \mathsf{log1p}\left(y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right)}, x, 0.5\right) \cdot x, x, x\right)\right) \cdot c \]

   7. Applied rewrites 86.3%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right) \cdot x, x, x\right)}\right) \cdot c \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.066:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-191}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right) \cdot x, x, x\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -235:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-191} \lor \neg \left(x \leq 4.2 \cdot 10^{-208}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -235.0)
   (* (* (expm1 x) y) c)
   (if (or (<= x -2.6e-191) (not (<= x 4.2e-208)))
     (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c)
     (* (* x c) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -235.0) {
		tmp = (expm1(x) * y) * c;
	} else if ((x <= -2.6e-191) || !(x <= 4.2e-208)) {
		tmp = log1p((y * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -235.0)
		tmp = Float64(Float64(expm1(x) * y) * c);
	elseif ((x <= -2.6e-191) || !(x <= 4.2e-208))
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -235.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], If[Or[LessEqual[x, -2.6e-191], N[Not[LessEqual[x, 4.2e-208]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -235

   1. Initial program 43.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.0

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 99.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 72.3

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

   7. Applied rewrites 72.3%

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

   if -235 < x < -2.59999999999999986e-191 or 4.20000000000000024e-208 < x

   1. Initial program 26.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.6

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 29.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 29.3

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 92.4

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

   4. Applied rewrites 92.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 88.9

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, x, 0.5\right)}, x, 1\right) \cdot x\right)\right) \cdot c \]

   7. Applied rewrites 88.9%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)}\right) \cdot c \]

   if -2.59999999999999986e-191 < x < 4.20000000000000024e-208

   1. Initial program 38.7%

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

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-commutative N/A

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

      16. *-lft-identity N/A

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

      17. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 95.8%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -235:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-191} \lor \neg \left(x \leq 4.2 \cdot 10^{-208}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0028:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-191} \lor \neg \left(x \leq 4.2 \cdot 10^{-208}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -0.0028)
   (* (* (expm1 x) y) c)
   (if (or (<= x -2.6e-191) (not (<= x 4.2e-208)))
     (* (log1p (* y (* (fma 0.5 x 1.0) x))) c)
     (* (* x c) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -0.0028) {
		tmp = (expm1(x) * y) * c;
	} else if ((x <= -2.6e-191) || !(x <= 4.2e-208)) {
		tmp = log1p((y * (fma(0.5, x, 1.0) * x))) * c;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -0.0028)
		tmp = Float64(Float64(expm1(x) * y) * c);
	elseif ((x <= -2.6e-191) || !(x <= 4.2e-208))
		tmp = Float64(log1p(Float64(y * Float64(fma(0.5, x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -0.0028], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], If[Or[LessEqual[x, -2.6e-191], N[Not[LessEqual[x, 4.2e-208]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00279999999999999997

   1. Initial program 43.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.8

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 71.2

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

   7. Applied rewrites 71.2%

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

   if -0.00279999999999999997 < x < -2.59999999999999986e-191 or 4.20000000000000024e-208 < x

   1. Initial program 25.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 25.5

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 27.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 27.3

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 89.7

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

   7. Applied rewrites 89.7%

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

   if -2.59999999999999986e-191 < x < 4.20000000000000024e-208

   1. Initial program 38.7%

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

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-commutative N/A

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

      16. *-lft-identity N/A

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

      17. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

   7. Applied rewrites 95.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0028:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-191} \lor \neg \left(x \leq 4.2 \cdot 10^{-208}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-18}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -3.6e-18) (* (* (expm1 x) y) c) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -3.6e-18) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -3.6e-18) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -3.6e-18:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -3.6e-18)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -3.6e-18], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6000000000000001e-18

   1. Initial program 43.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.2

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 97.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.4

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 70.7

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

   7. Applied rewrites 70.7%

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

   if -3.6000000000000001e-18 < x

   1. Initial program 30.0%

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

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-commutative N/A

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

      16. *-lft-identity N/A

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

      17. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 77.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-18}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 10^{-7}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1e-7) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e-7) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * 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 <= 1d-7) then
        tmp = (c * y) * x
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e-7) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1e-7:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1e-7)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1e-7)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1e-7], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 9.9999999999999995e-8

   1. Initial program 43.1%

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

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-commutative N/A

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

      16. *-lft-identity N/A

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

      17. lower-*.f64 49.9

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

   5. Applied rewrites 49.9%

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

   if 9.9999999999999995e-8 < c

   1. Initial program 17.9%

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

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-commutative N/A

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

      16. *-lft-identity N/A

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

      17. lower-*.f64 50.8

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

   5. Applied rewrites 50.8%

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

   7. Applied rewrites 60.3%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 10^{-7}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.2% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

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 * y) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.1%

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

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

   2. log-E N/A

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

   3. *-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-rgt-identity N/A

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

   8. metadata-eval N/A

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

   9. log-E N/A

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

   10. log-E N/A

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

   11. metadata-eval N/A

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

   12. log-E N/A

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

   13. lower-*.f64 N/A

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

   14. log-E N/A

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

   15. *-commutative N/A

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

   16. *-lft-identity N/A

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

   17. lower-*.f64 50.2

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

5. Applied rewrites 50.2%

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

6. Final simplification 50.2%

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

Developer Target 1: 93.6% 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 2025015 
(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)))))