Logarithmic Transform

Percentage Accurate: 41.0% → 99.2%

Time: 13.1s

Alternatives: 8

Speedup: 19.8×

• could not determine a ground truth

  1. c = 7.972138176877635e+99
  2. x = 3.5379765741934933e+220
  3. y = 3.991181432988946e-112

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: 41.0% 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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-7} \lor \neg \left(y \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1.6e-7) (not (<= y 2e-43)))
   (* (log1p (* y (expm1 x))) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.6e-7) || !(y <= 2e-43)) {
		tmp = log1p((y * expm1(x))) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.6e-7) || !(y <= 2e-43)) {
		tmp = Math.log1p((y * Math.expm1(x))) * c;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -1.6e-7) or not (y <= 2e-43):
		tmp = math.log1p((y * math.expm1(x))) * c
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1.6e-7) || !(y <= 2e-43))
		tmp = Float64(log1p(Float64(y * expm1(x))) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1.6e-7], N[Not[LessEqual[y, 2e-43]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6e-7 or 2.00000000000000015e-43 < y

   1. Initial program 33.2%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 33.2

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

      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 36.7

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

      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 36.7

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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 -1.6e-7 < y < 2.00000000000000015e-43

   1. Initial program 42.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 42.3

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

      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 67.2

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

      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 67.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-7} \lor \neg \left(y \leq 2 \cdot 10^{-43}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -105000:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -105000.0)
   (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c)
   (if (<= y 7.5e-20)
     (* (* (expm1 x) c) y)
     (*
      (log1p
       (*
        y
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -105000.0) {
		tmp = log1p((y * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c;
	} else if (y <= 7.5e-20) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = log1p((y * (fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x))) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -105000.0)
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c);
	elseif (y <= 7.5e-20)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x))) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -105000.0], N[(N[Log[1 + N[(y * N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 7.5e-20], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(y * N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -105000

   1. Initial program 44.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.0

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 44.0

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

      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 44.0

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

      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. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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%

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

   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 68.9

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

   7. Applied rewrites 68.9%

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

   if -105000 < y < 7.49999999999999981e-20

   1. Initial program 41.6%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 41.6

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

      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 67.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.4

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if 7.49999999999999981e-20 < y

   1. Initial program 17.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. 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 17.9

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

      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 17.9

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

      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 17.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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%

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

   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 97.9

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

   7. Applied rewrites 97.9%

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

Alternative 3: 89.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -105000 \lor \neg \left(y \leq 7.5 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -105000.0) (not (<= y 7.5e-20)))
   (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -105000.0) || !(y <= 7.5e-20)) {
		tmp = log1p((y * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -105000.0) || !(y <= 7.5e-20))
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -105000.0], N[Not[LessEqual[y, 7.5e-20]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -105000 or 7.49999999999999981e-20 < y

   1. Initial program 33.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 33.5

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

      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 33.5

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

      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 33.5

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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%

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

   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 80.6

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

   7. Applied rewrites 80.6%

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

   if -105000 < y < 7.49999999999999981e-20

   1. Initial program 41.6%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 41.6

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

      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 67.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 67.4

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 99.0

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

   7. Applied rewrites 99.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -105000 \lor \neg \left(y \leq 7.5 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+33} \lor \neg \left(y \leq 7.5 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -9.2e+33) (not (<= y 7.5e-20)))
   (* (log1p (* y (* (fma 0.5 x 1.0) x))) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -9.2e+33) || !(y <= 7.5e-20)) {
		tmp = log1p((y * (fma(0.5, x, 1.0) * x))) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -9.2e+33) || !(y <= 7.5e-20))
		tmp = Float64(log1p(Float64(y * Float64(fma(0.5, x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -9.2e+33], N[Not[LessEqual[y, 7.5e-20]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.20000000000000042e33 or 7.49999999999999981e-20 < y

   1. Initial program 31.6%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 31.6

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

      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 31.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.6

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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%

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

   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 80.8

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

   7. Applied rewrites 80.8%

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

   if -9.20000000000000042e33 < y < 7.49999999999999981e-20

   1. Initial program 42.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 42.3

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

      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 67.0

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

      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 67.0

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

      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. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 97.2

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

   7. Applied rewrites 97.2%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+33} \lor \neg \left(y \leq 7.5 \cdot 10^{-20}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+158} \lor \neg \left(y \leq 2.4 \cdot 10^{+198}\right):\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1.65e+158) (not (<= y 2.4e+198)))
   (* c (log (fma y x 1.0)))
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.65e+158) || !(y <= 2.4e+198)) {
		tmp = c * log(fma(y, x, 1.0));
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1.65e+158) || !(y <= 2.4e+198))
		tmp = Float64(c * log(fma(y, x, 1.0)));
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1.65e+158], N[Not[LessEqual[y, 2.4e+198]], $MachinePrecision]], N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65000000000000009e158 or 2.4000000000000001e198 < y

   1. Initial program 34.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \color{blue}{\left(1 + x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   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 67.9

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

   5. Applied rewrites 67.9%

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

   if -1.65000000000000009e158 < y < 2.4000000000000001e198

   1. Initial program 39.2%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 39.2

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

      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 58.2

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

      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 58.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      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 91.9

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

   4. Applied rewrites 91.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 89.1

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

   7. Applied rewrites 89.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+158} \lor \neg \left(y \leq 2.4 \cdot 10^{+198}\right):\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

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 53.6

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 53.6

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

   10. lift--.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow-to-exp N/A

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

   13. lift-E.f64 N/A

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

   14. log-E N/A

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

   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 93.4

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

4. Applied rewrites 93.4%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-expm1.f64 75.1

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

7. Applied rewrites 75.1%

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

Alternative 7: 62.8% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 6 \cdot 10^{-63}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 6e-63) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 6e-63) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * 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 <= 6d-63) then
        tmp = (c * y) * x
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 6e-63) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 6e-63:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 6e-63)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 6e-63)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 6e-63], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.99999999999999959e-63

   1. Initial program 44.2%

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-rgt-identity N/A

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

      16. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 5.99999999999999959e-63 < c

   1. Initial program 21.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-rgt-identity N/A

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

      16. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 52.6%

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

Alternative 8: 61.7% 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 38.3%

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

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-rgt-identity N/A

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

   8. metadata-eval N/A

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

   9. log-E N/A

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

   10. log-E N/A

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

   11. metadata-eval N/A

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

   12. log-E N/A

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

   13. lower-*.f64 N/A

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

   14. log-E N/A

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

   15. *-rgt-identity N/A

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

   16. lower-*.f64 59.8

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

5. Applied rewrites 59.8%

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

Developer Target 1: 92.9% 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 2024339 
(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)))))