Logarithmic Transform

Percentage Accurate: 41.5% → 99.1%

Time: 12.1s

Alternatives: 8

Speedup: 19.8×

• could not determine a ground truth

  1. c = 1.9662221870969775e-36
  2. x = 5.776926935062884e+80
  3. y = 1.1625299444064947e+56

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: 41.5% 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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-14} \lor \neg \left(y \leq 300000\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot y, -0.5, \mathsf{expm1}\left(x\right)\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1e-14) (not (<= y 300000.0)))
   (* (log1p (* y (expm1 x))) c)
   (* (* c (fma (* (pow (expm1 x) 2.0) y) -0.5 (expm1 x))) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1e-14) || !(y <= 300000.0)) {
		tmp = log1p((y * expm1(x))) * c;
	} else {
		tmp = (c * fma((pow(expm1(x), 2.0) * y), -0.5, expm1(x))) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1e-14) || !(y <= 300000.0))
		tmp = Float64(log1p(Float64(y * expm1(x))) * c);
	else
		tmp = Float64(Float64(c * fma(Float64((expm1(x) ^ 2.0) * y), -0.5, expm1(x))) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1e-14], N[Not[LessEqual[y, 300000.0]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * N[(N[(N[Power[N[(Exp[x] - 1), $MachinePrecision], 2.0], $MachinePrecision] * y), $MachinePrecision] * -0.5 + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999999e-15 or 3e5 < 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 36.3

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

      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 36.3

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

      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. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.99999999999999999e-15 < y < 3e5

   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 63.5

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

      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 63.5

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

      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. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 89.0

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

   4. Applied rewrites 89.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-14} \lor \neg \left(y \leq 300000\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot y, -0.5, \mathsf{expm1}\left(x\right)\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.0)
   (* c (* (expm1 x) y))
   (*
    (log1p
     (*
      y
      (*
       (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
       x)))
    c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.0

   1. Initial program 55.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. e-exp-1 N/A

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

      4. exp-prod N/A

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

      5. *-lft-identity N/A

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

      6. lower-expm1.f64 70.2

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

   8. Applied rewrites 70.2%

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

   if 0.0 < (pow.f64 (E.f64) x)

   1. Initial program 34.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 34.8

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

      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 35.9

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

      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 35.9

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

      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. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 90.9

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

   4. Applied rewrites 90.9%

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

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 90.8

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

   7. Applied rewrites 90.8%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.0)
   (* c (* (expm1 x) y))
   (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.0

   1. Initial program 55.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. e-exp-1 N/A

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

      4. exp-prod N/A

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

      5. *-lft-identity N/A

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

      6. lower-expm1.f64 70.2

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

   8. Applied rewrites 70.2%

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

   if 0.0 < (pow.f64 (E.f64) x)

   1. Initial program 34.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 34.8

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

      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 35.9

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

      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 35.9

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

      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. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 90.9

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

   4. Applied rewrites 90.9%

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

   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 90.6

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

   7. Applied rewrites 90.6%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.0)
   (* c (* (expm1 x) y))
   (* (log1p (* y (* (fma 0.5 x 1.0) x))) c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.0

   1. Initial program 55.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. e-exp-1 N/A

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

      4. exp-prod N/A

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

      5. *-lft-identity N/A

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

      6. lower-expm1.f64 70.2

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

   8. Applied rewrites 70.2%

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

   if 0.0 < (pow.f64 (E.f64) x)

   1. Initial program 34.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 34.8

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

      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 35.9

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

      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 35.9

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

      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. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 90.9

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

   4. Applied rewrites 90.9%

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

   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 90.5

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

   7. Applied rewrites 90.5%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.2%

   \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
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 40.2

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

   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 52.7

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

   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 52.7

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

   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. lift-E.f64 N/A

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

   13. e-exp-1 N/A

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

   14. pow-exp N/A

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

   15. *-lft-identity N/A

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

   16. lower-expm1.f64 93.3

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

4. Applied rewrites 93.3%

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

Alternative 6: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+220} \lor \neg \left(y \leq 1.35 \cdot 10^{+138}\right):\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -3.3e+220) (not (<= y 1.35e+138)))
   (* c (log (fma y x 1.0)))
   (* c (* (expm1 x) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -3.3e+220) || !(y <= 1.35e+138)) {
		tmp = c * log(fma(y, x, 1.0));
	} else {
		tmp = c * (expm1(x) * y);
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -3.3e+220) || !(y <= 1.35e+138))
		tmp = Float64(c * log(fma(y, x, 1.0)));
	else
		tmp = Float64(c * Float64(expm1(x) * y));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -3.3e+220], N[Not[LessEqual[y, 1.35e+138]], $MachinePrecision]], N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.30000000000000021e220 or 1.35000000000000004e138 < y

   1. Initial program 23.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 70.5

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

   5. Applied rewrites 70.5%

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

   if -3.30000000000000021e220 < y < 1.35000000000000004e138

   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. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. e-exp-1 N/A

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

      4. exp-prod N/A

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

      5. *-lft-identity N/A

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

      6. lower-expm1.f64 83.8

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

   8. Applied rewrites 83.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+220} \lor \neg \left(y \leq 1.35 \cdot 10^{+138}\right):\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-195}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot y\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -2.25e-195)
   (* c (* (expm1 x) y))
   (* (* c (* (fma 0.5 x 1.0) y)) x)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -2.25e-195) {
		tmp = c * (expm1(x) * y);
	} else {
		tmp = (c * (fma(0.5, x, 1.0) * y)) * x;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -2.25e-195)
		tmp = Float64(c * Float64(expm1(x) * y));
	else
		tmp = Float64(Float64(c * Float64(fma(0.5, x, 1.0) * y)) * x);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -2.25e-195], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(0.5 * x + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.25e-195

   1. Initial program 42.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. e-exp-1 N/A

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

      4. exp-prod N/A

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

      5. *-lft-identity N/A

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

      6. lower-expm1.f64 73.1

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

   8. Applied rewrites 73.1%

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

   if -2.25e-195 < x

   1. Initial program 38.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 38.3

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

      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 39.0

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

      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 39.0

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

      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. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 84.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-195}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot y\right)\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.3% 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

real(8) function code(c, x, y)
    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 40.2%

   \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
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. *-commutative N/A

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

   4. *-lft-identity N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. log-E N/A

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

   12. log-E N/A

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

   13. metadata-eval N/A

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

   14. log-E N/A

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

   15. lower-*.f64 N/A

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

   16. log-E N/A

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

   17. *-commutative N/A

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

   18. *-lft-identity N/A

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

   19. lower-*.f64 66.6

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

5. Applied rewrites 66.6%

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

Developer Target 1: 93.3% 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 2024298 
(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)))))