Logarithmic Transform

Percentage Accurate: 41.1% → 99.0%

Time: 11.1s

Alternatives: 7

Speedup: 19.8×

• could not determine a ground truth

  1. c = 4.006831908095774e-142
  2. x = 1.8902766593847494e+223
  3. y = 5.279563616986801e-19

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: 41.1% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-86} \lor \neg \left(y \leq 9.2 \cdot 10^{-79}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \frac{y}{\mathsf{fma}\left(1 + e^{x}, e^{x}, 1\right)}\right) \cdot \mathsf{expm1}\left(3 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -4.9e-86) (not (<= y 9.2e-79)))
   (* (log1p (* y (expm1 x))) c)
   (* (* c (/ y (fma (+ 1.0 (exp x)) (exp x) 1.0))) (expm1 (* 3.0 x)))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -4.9e-86) || !(y <= 9.2e-79)) {
		tmp = log1p((y * expm1(x))) * c;
	} else {
		tmp = (c * (y / fma((1.0 + exp(x)), exp(x), 1.0))) * expm1((3.0 * x));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -4.9e-86) || !(y <= 9.2e-79))
		tmp = Float64(log1p(Float64(y * expm1(x))) * c);
	else
		tmp = Float64(Float64(c * Float64(y / fma(Float64(1.0 + exp(x)), exp(x), 1.0))) * expm1(Float64(3.0 * x)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -4.9e-86], N[Not[LessEqual[y, 9.2e-79]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * N[(y / N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Exp[N[(3.0 * x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.89999999999999972e-86 or 9.20000000000000047e-79 < y

   1. Initial program 29.2%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 29.2

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

      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 39.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 39.1

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. 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%

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

   if -4.89999999999999972e-86 < y < 9.20000000000000047e-79

   1. Initial program 47.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 47.0

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

      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 68.0

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

      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 68.0

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

      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 84.3

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

   4. Applied rewrites 84.3%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 84.3

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

   7. Applied rewrites 84.3%

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

   9. Applied rewrites 99.8%

      \[\leadsto \frac{\mathsf{expm1}\left(x \cdot 3\right) \cdot \left(y \cdot c\right)}{\color{blue}{\mathsf{fma}\left(e^{x}, e^{x}, e^{x} + 1\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.8%

       \[\leadsto \left(c \cdot \frac{y}{\mathsf{fma}\left(1 + e^{x}, e^{x}, 1\right)}\right) \cdot \color{blue}{\mathsf{expm1}\left(3 \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{-86} \lor \neg \left(y \leq 9.2 \cdot 10^{-79}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \frac{y}{\mathsf{fma}\left(1 + e^{x}, e^{x}, 1\right)}\right) \cdot \mathsf{expm1}\left(3 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} \leq 0.9999998:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (pow (E) x) 0.9999998)
   (* (* (expm1 x) y) c)
   (* (log1p (* y (* (fma 0.5 x 1.0) x))) c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (E.f64) x) < 0.999999799999999994

   1. Initial program 49.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 49.4

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 99.4

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

      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 99.4

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 64.8

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

   7. Applied rewrites 64.8%

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

   if 0.999999799999999994 < (pow.f64 (E.f64) x)

   1. Initial program 32.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 32.0

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

      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 33.2

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

      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 33.2

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

      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 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 90.7

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

   7. Applied rewrites 90.7%

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

Alternative 3: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.7%

   \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
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 36.7

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

   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 51.3

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 51.3

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

   10. lift--.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow-to-exp N/A

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

   13. 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 93.2

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

4. Applied rewrites 93.2%

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

Alternative 4: 75.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+175} \lor \neg \left(y \leq 4.8 \cdot 10^{+69}\right):\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -2.3e+175) (not (<= y 4.8e+69)))
   (* c (log (fma y x 1.0)))
   (* (* (expm1 x) y) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -2.3e+175) || !(y <= 4.8e+69)) {
		tmp = c * log(fma(y, x, 1.0));
	} else {
		tmp = (expm1(x) * y) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -2.3e+175) || !(y <= 4.8e+69))
		tmp = Float64(c * log(fma(y, x, 1.0)));
	else
		tmp = Float64(Float64(expm1(x) * y) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -2.3e+175], N[Not[LessEqual[y, 4.8e+69]], $MachinePrecision]], N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3e175 or 4.8000000000000003e69 < y

   1. Initial program 31.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if -2.3e175 < y < 4.8000000000000003e69

   1. Initial program 38.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 38.0

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

      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 56.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 56.3

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. 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 91.5

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

   4. Applied rewrites 91.5%

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

   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

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 84.3

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

   7. Applied rewrites 84.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+175} \lor \neg \left(y \leq 4.8 \cdot 10^{+69}\right):\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-79}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -1.4e-79) (* (* (expm1 x) y) c) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -1.4e-79) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -1.4e-79) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -1.4e-79:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -1.4e-79)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -1.4e-79], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000006e-79

   1. Initial program 43.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 43.8

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 84.2

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

      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 84.2

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

      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 99.8

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

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-expm1.f64 68.2

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

   7. Applied rewrites 68.2%

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

   if -1.40000000000000006e-79 < x

   1. Initial program 33.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-rgt-identity N/A

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

      16. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 79.7%

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

Alternative 6: 62.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 4.5 \cdot 10^{+134}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 4.5e+134) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 4.5e+134) {
		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 <= 4.5d+134) 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 <= 4.5e+134) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 4.5e+134:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 4.5e+134)
		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 <= 4.5e+134)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 4.5e+134], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 4.4999999999999997e134

   1. Initial program 41.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-rgt-identity N/A

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

      16. lower-*.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if 4.4999999999999997e134 < c

   1. Initial program 10.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. log-E N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. lower-*.f64 N/A

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

      14. log-E N/A

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

      15. *-rgt-identity N/A

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

      16. lower-*.f64 31.7

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

   5. Applied rewrites 31.7%

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

   7. Applied rewrites 50.2%

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

Alternative 7: 60.7% 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 36.7%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. log-E N/A

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

   3. *-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-rgt-identity N/A

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

   8. metadata-eval N/A

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

   9. log-E N/A

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

   10. log-E N/A

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

   11. metadata-eval N/A

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

   12. log-E N/A

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

   13. lower-*.f64 N/A

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

   14. log-E N/A

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

   15. *-rgt-identity N/A

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

   16. lower-*.f64 60.7

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

5. Applied rewrites 60.7%

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

Developer Target 1: 93.8% 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 2024320 
(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)))))