Logarithmic Transform

Percentage Accurate: 40.9% → 91.9%

Time: 13.3s

Alternatives: 7

Speedup: 19.8×

• could not determine a ground truth

  1. c = 4.490958185088388e+287
  2. x = 4.1246901882193196e+239
  3. y = 1.052634823146021e+82

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: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;x \leq 5.9 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= x 5.9e-305)
     t_0
     (if (<= x 1.7e-76)
       (*
        (*
         (fma
          (fma
           (* 0.3333333333333333 c)
           (* (* x x) y)
           (* (* (fma x c c) -0.5) x))
          y
          (fma (fma 0.16666666666666666 (* c x) (* 0.5 c)) x c))
         x)
        y)
       t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (x <= 5.9e-305) {
		tmp = t_0;
	} else if (x <= 1.7e-76) {
		tmp = (fma(fma((0.3333333333333333 * c), ((x * x) * y), ((fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, (c * x), (0.5 * c)), x, c)) * x) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(expm1(x) * y)))
	tmp = 0.0
	if (x <= 5.9e-305)
		tmp = t_0;
	elseif (x <= 1.7e-76)
		tmp = Float64(Float64(fma(fma(Float64(0.3333333333333333 * c), Float64(Float64(x * x) * y), Float64(Float64(fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, Float64(c * x), Float64(0.5 * c)), x, c)) * x) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.9e-305], t$95$0, If[LessEqual[x, 1.7e-76], N[(N[(N[(N[(N[(0.3333333333333333 * c), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(x * c + c), $MachinePrecision] * -0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[(0.16666666666666666 * N[(c * x), $MachinePrecision] + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * x + c), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.9000000000000005e-305 or 1.7e-76 < x

   1. Initial program 34.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 34.4

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

      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 54.9

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

      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 54.9

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

      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 94.3

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

   4. Applied rewrites 94.3%

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

   if 5.9000000000000005e-305 < x < 1.7e-76

   1. Initial program 30.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot \mathsf{fma}\left(0.3333333333333333 \cdot y, {\mathsf{E}\left(\right)}^{x} - 1, -0.5\right)\right), y, \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot c\right) \cdot y} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c, \left(\mathsf{fma}\left(0.3333333333333333, y, -0.5\right) \cdot y\right) \cdot c\right), x, \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(y \cdot c\right)\right)\right), x, c\right) \cdot x\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(c + \left(x \cdot \left(\frac{1}{6} \cdot \left(c \cdot x\right) + \frac{1}{2} \cdot c\right) + y \cdot \left(\frac{1}{3} \cdot \left(c \cdot \left({x}^{2} \cdot y\right)\right) + x \cdot \left(\frac{-1}{2} \cdot c + \frac{-1}{2} \cdot \left(c \cdot x\right)\right)\right)\right)\right) \cdot x\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(-0.5 \cdot \mathsf{fma}\left(x, c, c\right)\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot c, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.9 \cdot 10^{-305}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;x \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c)))
   (if (<= x -8e-5)
     (* (* (expm1 x) y) c)
     (if (<= x 5.9e-305)
       t_0
       (if (<= x 1.7e-76)
         (*
          (*
           (fma
            (fma
             (* 0.3333333333333333 c)
             (* (* x x) y)
             (* (* (fma x c c) -0.5) x))
            y
            (fma (fma 0.16666666666666666 (* c x) (* 0.5 c)) x c))
           x)
          y)
         t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = log1p(((fma(0.5, x, 1.0) * x) * y)) * c;
	double tmp;
	if (x <= -8e-5) {
		tmp = (expm1(x) * y) * c;
	} else if (x <= 5.9e-305) {
		tmp = t_0;
	} else if (x <= 1.7e-76) {
		tmp = (fma(fma((0.3333333333333333 * c), ((x * x) * y), ((fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, (c * x), (0.5 * c)), x, c)) * x) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(Float64(fma(0.5, x, 1.0) * x) * y)) * c)
	tmp = 0.0
	if (x <= -8e-5)
		tmp = Float64(Float64(expm1(x) * y) * c);
	elseif (x <= 5.9e-305)
		tmp = t_0;
	elseif (x <= 1.7e-76)
		tmp = Float64(Float64(fma(fma(Float64(0.3333333333333333 * c), Float64(Float64(x * x) * y), Float64(Float64(fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, Float64(c * x), Float64(0.5 * c)), x, c)) * x) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[x, -8e-5], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 5.9e-305], t$95$0, If[LessEqual[x, 1.7e-76], N[(N[(N[(N[(N[(0.3333333333333333 * c), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(x * c + c), $MachinePrecision] * -0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[(0.16666666666666666 * N[(c * x), $MachinePrecision] + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * x + c), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.00000000000000065e-5

   1. Initial program 48.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 48.5

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

      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 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. 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 99.8

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

   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 67.9

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

   7. Applied rewrites 67.9%

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

   if -8.00000000000000065e-5 < x < 5.9000000000000005e-305 or 1.7e-76 < x

   1. Initial program 25.2%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 25.2

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

      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 25.4

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

      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 25.4

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

      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 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 90.2

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

   7. Applied rewrites 90.2%

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

   if 5.9000000000000005e-305 < x < 1.7e-76

   1. Initial program 30.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot \mathsf{fma}\left(0.3333333333333333 \cdot y, {\mathsf{E}\left(\right)}^{x} - 1, -0.5\right)\right), y, \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot c\right) \cdot y} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c, \left(\mathsf{fma}\left(0.3333333333333333, y, -0.5\right) \cdot y\right) \cdot c\right), x, \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(y \cdot c\right)\right)\right), x, c\right) \cdot x\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(c + \left(x \cdot \left(\frac{1}{6} \cdot \left(c \cdot x\right) + \frac{1}{2} \cdot c\right) + y \cdot \left(\frac{1}{3} \cdot \left(c \cdot \left({x}^{2} \cdot y\right)\right) + x \cdot \left(\frac{-1}{2} \cdot c + \frac{-1}{2} \cdot \left(c \cdot x\right)\right)\right)\right)\right) \cdot x\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(-0.5 \cdot \mathsf{fma}\left(x, c, c\right)\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot c, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* y x)) c)))
   (if (<= x -3.6e-15)
     (* (* (expm1 x) y) c)
     (if (<= x 5.9e-305)
       t_0
       (if (<= x 1.7e-76)
         (*
          (*
           (fma
            (fma
             (* 0.3333333333333333 c)
             (* (* x x) y)
             (* (* (fma x c c) -0.5) x))
            y
            (fma (fma 0.16666666666666666 (* c x) (* 0.5 c)) x c))
           x)
          y)
         t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = log1p((y * x)) * c;
	double tmp;
	if (x <= -3.6e-15) {
		tmp = (expm1(x) * y) * c;
	} else if (x <= 5.9e-305) {
		tmp = t_0;
	} else if (x <= 1.7e-76) {
		tmp = (fma(fma((0.3333333333333333 * c), ((x * x) * y), ((fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, (c * x), (0.5 * c)), x, c)) * x) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(y * x)) * c)
	tmp = 0.0
	if (x <= -3.6e-15)
		tmp = Float64(Float64(expm1(x) * y) * c);
	elseif (x <= 5.9e-305)
		tmp = t_0;
	elseif (x <= 1.7e-76)
		tmp = Float64(Float64(fma(fma(Float64(0.3333333333333333 * c), Float64(Float64(x * x) * y), Float64(Float64(fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, Float64(c * x), Float64(0.5 * c)), x, c)) * x) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[x, -3.6e-15], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[x, 5.9e-305], t$95$0, If[LessEqual[x, 1.7e-76], N[(N[(N[(N[(N[(0.3333333333333333 * c), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(x * c + c), $MachinePrecision] * -0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[(0.16666666666666666 * N[(c * x), $MachinePrecision] + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * x + c), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6000000000000001e-15

   1. Initial program 48.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 48.5

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

      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 98.9

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

      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 98.9

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

      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 99.8

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

   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 68.6

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

   7. Applied rewrites 68.6%

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

   if -3.6000000000000001e-15 < x < 5.9000000000000005e-305 or 1.7e-76 < x

   1. Initial program 24.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 24.7

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

      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 24.7

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

      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 24.7

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

      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 90.5

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

   4. Applied rewrites 90.5%

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

   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 90.1

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

   7. Applied rewrites 90.1%

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

   if 5.9000000000000005e-305 < x < 1.7e-76

   1. Initial program 30.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot \mathsf{fma}\left(0.3333333333333333 \cdot y, {\mathsf{E}\left(\right)}^{x} - 1, -0.5\right)\right), y, \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot c\right) \cdot y} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 79.5%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c, \left(\mathsf{fma}\left(0.3333333333333333, y, -0.5\right) \cdot y\right) \cdot c\right), x, \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(y \cdot c\right)\right)\right), x, c\right) \cdot x\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(c + \left(x \cdot \left(\frac{1}{6} \cdot \left(c \cdot x\right) + \frac{1}{2} \cdot c\right) + y \cdot \left(\frac{1}{3} \cdot \left(c \cdot \left({x}^{2} \cdot y\right)\right) + x \cdot \left(\frac{-1}{2} \cdot c + \frac{-1}{2} \cdot \left(c \cdot x\right)\right)\right)\right)\right) \cdot x\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 94.5%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(-0.5 \cdot \mathsf{fma}\left(x, c, c\right)\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot c, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-15}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-305}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-29}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -6.8e-29)
   (* (* (expm1 x) y) c)
   (*
    (*
     (fma
      (fma (* 0.3333333333333333 c) (* (* x x) y) (* (* (fma x c c) -0.5) x))
      y
      (fma (fma 0.16666666666666666 (* c x) (* 0.5 c)) x c))
     x)
    y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -6.8e-29) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (fma(fma((0.3333333333333333 * c), ((x * x) * y), ((fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, (c * x), (0.5 * c)), x, c)) * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -6.8e-29)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(fma(fma(Float64(0.3333333333333333 * c), Float64(Float64(x * x) * y), Float64(Float64(fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, Float64(c * x), Float64(0.5 * c)), x, c)) * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -6.8e-29], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 * c), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(x * c + c), $MachinePrecision] * -0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[(0.16666666666666666 * N[(c * x), $MachinePrecision] + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * x + c), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.79999999999999945e-29

   1. Initial program 45.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.8

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 92.2

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

      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 92.2

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   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 66.9

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

   7. Applied rewrites 66.9%

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

   if -6.79999999999999945e-29 < x

   1. Initial program 27.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot \mathsf{fma}\left(0.3333333333333333 \cdot y, {\mathsf{E}\left(\right)}^{x} - 1, -0.5\right)\right), y, \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot c\right) \cdot y} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.2%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c, \left(\mathsf{fma}\left(0.3333333333333333, y, -0.5\right) \cdot y\right) \cdot c\right), x, \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(y \cdot c\right)\right)\right), x, c\right) \cdot x\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(c + \left(x \cdot \left(\frac{1}{6} \cdot \left(c \cdot x\right) + \frac{1}{2} \cdot c\right) + y \cdot \left(\frac{1}{3} \cdot \left(c \cdot \left({x}^{2} \cdot y\right)\right) + x \cdot \left(\frac{-1}{2} \cdot c + \frac{-1}{2} \cdot \left(c \cdot x\right)\right)\right)\right)\right) \cdot x\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 81.8%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(-0.5 \cdot \mathsf{fma}\left(x, c, c\right)\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot c, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-29}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 8.5 \cdot 10^{+63}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 8.5e+63)
   (* (* c y) x)
   (*
    (*
     (fma
      (fma (* 0.3333333333333333 c) (* (* x x) y) (* (* (fma x c c) -0.5) x))
      y
      (fma (fma 0.16666666666666666 (* c x) (* 0.5 c)) x c))
     x)
    y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 8.5e+63) {
		tmp = (c * y) * x;
	} else {
		tmp = (fma(fma((0.3333333333333333 * c), ((x * x) * y), ((fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, (c * x), (0.5 * c)), x, c)) * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 8.5e+63)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(fma(fma(Float64(0.3333333333333333 * c), Float64(Float64(x * x) * y), Float64(Float64(fma(x, c, c) * -0.5) * x)), y, fma(fma(0.16666666666666666, Float64(c * x), Float64(0.5 * c)), x, c)) * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 8.5e+63], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(0.3333333333333333 * c), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] + N[(N[(N[(x * c + c), $MachinePrecision] * -0.5), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * y + N[(N[(0.16666666666666666 * N[(c * x), $MachinePrecision] + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * x + c), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 8.5000000000000004e63

   1. Initial program 40.5%

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

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

      14. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

   if 8.5000000000000004e63 < c

   1. Initial program 14.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c \cdot \left({\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2} \cdot \mathsf{fma}\left(0.3333333333333333 \cdot y, {\mathsf{E}\left(\right)}^{x} - 1, -0.5\right)\right), y, \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot c\right) \cdot y} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.6%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c, \left(\mathsf{fma}\left(0.3333333333333333, y, -0.5\right) \cdot y\right) \cdot c\right), x, \mathsf{fma}\left(0.5, c, -0.5 \cdot \left(y \cdot c\right)\right)\right), x, c\right) \cdot x\right) \cdot y \]

   9. Taylor expanded in y around 0

      \[\leadsto \left(\left(c + \left(x \cdot \left(\frac{1}{6} \cdot \left(c \cdot x\right) + \frac{1}{2} \cdot c\right) + y \cdot \left(\frac{1}{3} \cdot \left(c \cdot \left({x}^{2} \cdot y\right)\right) + x \cdot \left(\frac{-1}{2} \cdot c + \frac{-1}{2} \cdot \left(c \cdot x\right)\right)\right)\right)\right) \cdot x\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 58.2%

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(-0.5 \cdot \mathsf{fma}\left(x, c, c\right)\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot c, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 8.5 \cdot 10^{+63}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333 \cdot c, \left(x \cdot x\right) \cdot y, \left(\mathsf{fma}\left(x, c, c\right) \cdot -0.5\right) \cdot x\right), y, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, c \cdot x, 0.5 \cdot c\right), x, c\right)\right) \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.16 \cdot 10^{-66}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.16e-66) (* (* c y) x) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.16e-66) {
		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 <= 1.16d-66) 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 <= 1.16e-66) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1.16e-66:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.16e-66)
		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 <= 1.16e-66)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.16e-66], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.16000000000000002e-66

   1. Initial program 41.2%

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

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

      14. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if 1.16000000000000002e-66 < c

   1. Initial program 18.6%

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

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

      14. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 57.1%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.16 \cdot 10^{-66}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 33.4%

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

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

   14. lower-*.f64 55.9

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

5. Applied rewrites 55.9%

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

7. Applied rewrites 57.2%

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

Developer Target 1: 93.5% 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 2024264 
(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)))))