Logarithmic Transform

Percentage Accurate: 40.9% → 98.5%

Time: 14.2s

Alternatives: 9

Speedup: 19.8×

• could not determine a ground truth

  1. c = 4.996360554386734e-308
  2. x = 5.48325423410749e+19
  3. y = 2.131538570684928e-290

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: 40.9% 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: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+29} \lor \neg \left(y \leq 1300000000\right):\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5 \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right) \cdot y, c, c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1.22e+29) (not (<= y 1300000000.0)))
   (* c (log1p (* (expm1 x) y)))
   (* (fma (* (* -0.5 (pow (expm1 x) 2.0)) y) c (* c (expm1 x))) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.22e+29) || !(y <= 1300000000.0)) {
		tmp = c * log1p((expm1(x) * y));
	} else {
		tmp = fma(((-0.5 * pow(expm1(x), 2.0)) * y), c, (c * expm1(x))) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1.22e+29) || !(y <= 1300000000.0))
		tmp = Float64(c * log1p(Float64(expm1(x) * y)));
	else
		tmp = Float64(fma(Float64(Float64(-0.5 * (expm1(x) ^ 2.0)) * y), c, Float64(c * expm1(x))) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1.22e+29], N[Not[LessEqual[y, 1300000000.0]], $MachinePrecision]], N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-0.5 * N[Power[N[(Exp[x] - 1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * c + N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.22e29 or 1.3e9 < y

   1. Initial program 33.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 33.9

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

      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 33.9

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

      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 33.9

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

      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 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 -1.22e29 < y < 1.3e9

   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. 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 42.5

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

      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 72.3

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

      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 72.3

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

      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 88.4

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

   4. Applied rewrites 88.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+29} \lor \neg \left(y \leq 1300000000\right):\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-0.5 \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right) \cdot y, c, c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.8:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.8)
   (* (* (expm1 x) y) c)
   (*
    (log1p
     (*
      (*
       (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
       x)
      y))
    c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.80000000000000004

   1. Initial program 46.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 46.4

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

      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. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if 0.80000000000000004 < (pow.f64 (E.f64) x)

   1. Initial program 35.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 35.7

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

      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 37.1

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

      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 37.1

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

      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 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 88.6

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

   7. Applied rewrites 88.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.8:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.8:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.8)
   (* (* (expm1 x) y) c)
   (* (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y)) c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.80000000000000004

   1. Initial program 46.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 46.4

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

      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. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if 0.80000000000000004 < (pow.f64 (E.f64) x)

   1. Initial program 35.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 35.7

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

      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 37.1

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

      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 37.1

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

      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 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 88.6

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

   7. Applied rewrites 88.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.8:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.8:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.8)
   (* (* (expm1 x) y) c)
   (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.80000000000000004

   1. Initial program 46.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 46.4

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

      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. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if 0.80000000000000004 < (pow.f64 (E.f64) x)

   1. Initial program 35.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 35.7

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

      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 37.1

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

      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 37.1

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

      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 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 88.4

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

   7. Applied rewrites 88.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.8:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.8:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.8) (* (* (expm1 x) y) c) (* (log1p (* x y)) c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.80000000000000004

   1. Initial program 46.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 46.4

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

      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. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if 0.80000000000000004 < (pow.f64 (E.f64) x)

   1. Initial program 35.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 35.7

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

      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 37.1

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

      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 37.1

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

      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 x around 0

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

      1. lower-*.f64 87.6

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

   7. Applied rewrites 87.6%

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

Alternative 6: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.999999999999998:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.999999999999998) (* (* (expm1 x) y) c) (* (* c y) x)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.999999999999998

   1. Initial program 45.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 45.9

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

      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 97.2

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

      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 97.2

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

      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. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.0

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

   7. Applied rewrites 71.0%

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

   if 0.999999999999998 < (pow.f64 (E.f64) x)

   1. Initial program 35.7%

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.999999999999998:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.2% 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]

Derivation

1. Initial program 39.3%

   \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
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 39.3

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

   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 57.9

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

   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 57.9

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

   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 92.6

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

4. Applied rewrites 92.6%

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

5. Final simplification 92.6%

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

Alternative 8: 63.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.2 \cdot 10^{-30}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.2e-30) (* (* c y) x) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.2e-30) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(c, x, y)
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 1.2d-30) then
        tmp = (c * y) * x
    else
        tmp = (c * x) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.2e-30) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1.2e-30:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.2e-30)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1.2e-30)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.2e-30], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.19999999999999992e-30

   1. Initial program 49.9%

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if 1.19999999999999992e-30 < c

   1. Initial program 13.7%

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. lower-*.f64 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 58.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.2 \cdot 10^{-30}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.3% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. log-E N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. log-E N/A

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

   13. *-commutative N/A

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

   14. *-lft-identity N/A

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

   15. lower-*.f64 58.8

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

5. Applied rewrites 58.8%

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

7. Applied rewrites 56.3%

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

Developer Target 1: 93.2% 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 2024271 
(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)))))