Logarithmic Transform

Percentage Accurate: 42.4% → 99.2%

Time: 13.2s

Alternatives: 13

Speedup: 19.8×

• could not determine a ground truth

  1. c = -1.915969181407994e+303
  2. x = 5.541633966260884e+105
  3. y = 5.2199825196387e+288

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: 42.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(3 \cdot x\right) \cdot c}{\mathsf{fma}\left(e^{x}, e^{x}, e^{x} + 1\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= y -7e-59)
     t_0
     (if (<= y 9e-34)
       (* (/ (* (expm1 (* 3.0 x)) c) (fma (exp x) (exp x) (+ (exp x) 1.0))) y)
       t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -7e-59) {
		tmp = t_0;
	} else if (y <= 9e-34) {
		tmp = ((expm1((3.0 * x)) * c) / fma(exp(x), exp(x), (exp(x) + 1.0))) * 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 (y <= -7e-59)
		tmp = t_0;
	elseif (y <= 9e-34)
		tmp = Float64(Float64(Float64(expm1(Float64(3.0 * x)) * c) / fma(exp(x), exp(x), Float64(exp(x) + 1.0))) * 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[y, -7e-59], t$95$0, If[LessEqual[y, 9e-34], N[(N[(N[(N[(Exp[N[(3.0 * x), $MachinePrecision]] - 1), $MachinePrecision] * c), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] * N[Exp[x], $MachinePrecision] + N[(N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.0000000000000002e-59 or 9.00000000000000085e-34 < y

   1. Initial program 35.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 35.1

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 37.3

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

      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 37.3

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

      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 98.9

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

   4. Applied rewrites 98.9%

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

   if -7.0000000000000002e-59 < y < 9.00000000000000085e-34

   1. Initial program 44.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-E.f64 67.1

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

   5. Applied rewrites 67.1%

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

   7. Applied rewrites 67.2%

      \[\leadsto \left(\left({\left(e^{3}\right)}^{\left(0.3333333333333333 \cdot x\right)} - 1\right) \cdot c\right) \cdot y \]

   9. Applied rewrites 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-59}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(3 \cdot x\right) \cdot c}{\mathsf{fma}\left(e^{x}, e^{x}, e^{x} + 1\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 39.3

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

   4. lift-log.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lower-log1p.f64 51.4

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

   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 51.4

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

   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 93.6

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

4. Applied rewrites 93.6%

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

5. Final simplification 93.6%

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

Alternative 3: 86.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.39:\\ \;\;\;\;\mathsf{log1p}\left(\frac{y}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}\right) \cdot c\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+41}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), x, -0.5\right), x, 1\right)}{x}}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -0.39)
   (* (log1p (/ y (/ (fma -0.5 x 1.0) x))) c)
   (if (<= y 1.28e+41)
     (* (* (expm1 x) y) c)
     (*
      (log1p
       (/
        y
        (/
         (fma
          (fma (fma -0.001388888888888889 (* x x) 0.08333333333333333) x -0.5)
          x
          1.0)
         x)))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -0.39) {
		tmp = log1p((y / (fma(-0.5, x, 1.0) / x))) * c;
	} else if (y <= 1.28e+41) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p((y / (fma(fma(fma(-0.001388888888888889, (x * x), 0.08333333333333333), x, -0.5), x, 1.0) / x))) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -0.39)
		tmp = Float64(log1p(Float64(y / Float64(fma(-0.5, x, 1.0) / x))) * c);
	elseif (y <= 1.28e+41)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(y / Float64(fma(fma(fma(-0.001388888888888889, Float64(x * x), 0.08333333333333333), x, -0.5), x, 1.0) / x))) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -0.39], N[(N[Log[1 + N[(y / N[(N[(-0.5 * x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.28e+41], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(y / N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * x + -0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.39000000000000001

   1. Initial program 55.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 55.0

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

      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 55.0

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

      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 55.0

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-expm1.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow2 N/A

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

      9. pow-exp N/A

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

      10. metadata-eval N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\color{blue}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

      12. lower-*.f64 99.5

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(\color{blue}{x \cdot 2}\right)}}\right) \cdot c \]

   6. Applied rewrites 99.5%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 70.4

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

   9. Applied rewrites 70.4%

      \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}}\right) \cdot c \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. un-div-inv N/A

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

       4. lower-/.f64 70.4

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

   11. Applied rewrites 70.4%

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

   if -0.39000000000000001 < y < 1.27999999999999992e41

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.2

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

      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 62.0

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

      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 62.0

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 89.3

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

   4. Applied rewrites 89.3%

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

   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 88.8

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

   7. Applied rewrites 88.8%

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

   if 1.27999999999999992e41 < y

   1. Initial program 12.2%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 12.2

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

      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 12.2

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

      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 12.2

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

      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 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-expm1.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow2 N/A

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

      9. pow-exp N/A

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

      10. metadata-eval N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\color{blue}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

      12. lower-*.f64 99.5

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(\color{blue}{x \cdot 2}\right)}}\right) \cdot c \]

   6. Applied rewrites 99.5%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{1 + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}{x}}}\right) \cdot c \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{1 + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)}{x}}}\right) \cdot c \]

      2. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1}}{x}}\right) \cdot c \]

      3. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\color{blue}{\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) \cdot x} + 1}{x}}\right) \cdot c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, x, 1\right)}}{x}}\right) \cdot c \]

      5. sub-neg N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, x, 1\right)}{x}}\right) \cdot c \]

      6. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), x, 1\right)}{x}}\right) \cdot c \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x + \color{blue}{\frac{-1}{2}}, x, 1\right)}{x}}\right) \cdot c \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}, x, \frac{-1}{2}\right)}, x, 1\right)}{x}}\right) \cdot c \]

      9. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{720} \cdot {x}^{2} + \frac{1}{12}}, x, \frac{-1}{2}\right), x, 1\right)}{x}}\right) \cdot c \]

      10. *-commutative N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{12}, x, \frac{-1}{2}\right), x, 1\right)}{x}}\right) \cdot c \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{12}\right)}, x, \frac{-1}{2}\right), x, 1\right)}{x}}\right) \cdot c \]

      12. unpow2 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{-1}{2}\right), x, 1\right)}{x}}\right) \cdot c \]

      13. lower-*.f64 99.5

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.08333333333333333\right), x, -0.5\right), x, 1\right)}{x}}\right) \cdot c \]

   9. Applied rewrites 99.5%

      \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, -0.5\right), x, 1\right)}{x}}}\right) \cdot c \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{log1p}\left(\color{blue}{y \cdot \frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{-1}{2}\right), x, 1\right)}{x}}}\right) \cdot c \]

       2. lift-/.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{-1}{2}\right), x, 1\right)}{x}}}\right) \cdot c \]

       3. un-div-inv N/A

          \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{720}, \frac{1}{12}\right), x, \frac{-1}{2}\right), x, 1\right)}{x}}}\right) \cdot c \]

       4. lower-/.f64 99.6

          \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, -0.5\right), x, 1\right)}{x}}}\right) \cdot c \]

   11. Applied rewrites 99.6%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), x, -0.5\right), x, 1\right)}{x}}}\right) \cdot c \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.39:\\ \;\;\;\;\mathsf{log1p}\left(\frac{y}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}\right) \cdot c\\ \mathbf{elif}\;y \leq 5:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, 1\right)}{x}} \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -0.39)
   (* (log1p (/ y (/ (fma -0.5 x 1.0) x))) c)
   (if (<= y 5.0)
     (* (* (expm1 x) y) c)
     (*
      (log1p (* (/ 1.0 (/ (fma (fma 0.08333333333333333 x -0.5) x 1.0) x)) y))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -0.39) {
		tmp = log1p((y / (fma(-0.5, x, 1.0) / x))) * c;
	} else if (y <= 5.0) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p(((1.0 / (fma(fma(0.08333333333333333, x, -0.5), x, 1.0) / x)) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -0.39)
		tmp = Float64(log1p(Float64(y / Float64(fma(-0.5, x, 1.0) / x))) * c);
	elseif (y <= 5.0)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(Float64(1.0 / Float64(fma(fma(0.08333333333333333, x, -0.5), x, 1.0) / x)) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -0.39], N[(N[Log[1 + N[(y / N[(N[(-0.5 * x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 5.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(1.0 / N[(N[(N[(0.08333333333333333 * x + -0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.39000000000000001

   1. Initial program 55.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 55.0

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

      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 55.0

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

      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 55.0

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-expm1.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow2 N/A

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

      9. pow-exp N/A

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

      10. metadata-eval N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\color{blue}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

      12. lower-*.f64 99.5

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(\color{blue}{x \cdot 2}\right)}}\right) \cdot c \]

   6. Applied rewrites 99.5%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 70.4

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

   9. Applied rewrites 70.4%

      \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}}\right) \cdot c \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. un-div-inv N/A

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

       4. lower-/.f64 70.4

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

   11. Applied rewrites 70.4%

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

   if -0.39000000000000001 < y < 5

   1. Initial program 42.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.3

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 64.4

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

      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 64.4

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 89.3

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

   4. Applied rewrites 89.3%

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

   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 88.8

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

   7. Applied rewrites 88.8%

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

   if 5 < y

   1. Initial program 14.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 14.3

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 14.3

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

      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 14.3

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

      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 97.8

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

   4. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-expm1.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow2 N/A

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

      9. pow-exp N/A

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

      10. metadata-eval N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\color{blue}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

      12. lower-*.f64 97.7

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(\color{blue}{x \cdot 2}\right)}}\right) \cdot c \]

   6. Applied rewrites 97.7%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{1 + x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)}{x}}}\right) \cdot c \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{1 + x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right)}{x}}}\right) \cdot c \]

      2. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\color{blue}{x \cdot \left(\frac{1}{12} \cdot x - \frac{1}{2}\right) + 1}}{x}}\right) \cdot c \]

      3. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\color{blue}{\left(\frac{1}{12} \cdot x - \frac{1}{2}\right) \cdot x} + 1}{x}}\right) \cdot c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{1}{12} \cdot x - \frac{1}{2}, x, 1\right)}}{x}}\right) \cdot c \]

      5. sub-neg N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\frac{1}{12} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, x, 1\right)}{x}}\right) \cdot c \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\frac{1}{12} \cdot x + \color{blue}{\frac{-1}{2}}, x, 1\right)}{x}}\right) \cdot c \]

      7. lower-fma.f64 97.6

         \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.08333333333333333, x, -0.5\right)}, x, 1\right)}{x}}\right) \cdot c \]

   9. Applied rewrites 97.6%

      \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, 1\right)}{x}}}\right) \cdot c \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.39:\\ \;\;\;\;\mathsf{log1p}\left(\frac{y}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}\right) \cdot c\\ \mathbf{elif}\;y \leq 5:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, 1\right)}{x}} \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.39:\\ \;\;\;\;\mathsf{log1p}\left(\frac{y}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}\right) \cdot c\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -0.39)
   (* (log1p (/ y (/ (fma -0.5 x 1.0) x))) c)
   (if (<= y 0.2)
     (* (* (expm1 x) y) c)
     (*
      (log1p
       (*
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)
        y))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -0.39) {
		tmp = log1p((y / (fma(-0.5, x, 1.0) / x))) * c;
	} else if (y <= 0.2) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -0.39)
		tmp = Float64(log1p(Float64(y / Float64(fma(-0.5, x, 1.0) / x))) * c);
	elseif (y <= 0.2)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -0.39], N[(N[Log[1 + N[(y / N[(N[(-0.5 * x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.2], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.39000000000000001

   1. Initial program 55.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 55.0

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

      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 55.0

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

      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 55.0

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-expm1.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow2 N/A

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

      9. pow-exp N/A

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

      10. metadata-eval N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\color{blue}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

      12. lower-*.f64 99.5

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(\color{blue}{x \cdot 2}\right)}}\right) \cdot c \]

   6. Applied rewrites 99.5%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 70.4

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

   9. Applied rewrites 70.4%

      \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}}\right) \cdot c \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. un-div-inv N/A

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

       4. lower-/.f64 70.4

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

   11. Applied rewrites 70.4%

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

   if -0.39000000000000001 < y < 0.20000000000000001

   1. Initial program 42.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.3

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 64.4

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

      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 64.4

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 89.3

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

   4. Applied rewrites 89.3%

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

   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 88.8

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

   7. Applied rewrites 88.8%

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

   if 0.20000000000000001 < y

   1. Initial program 14.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 14.3

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 14.3

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

      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 14.3

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

      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 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

   7. Applied rewrites 97.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.39:\\ \;\;\;\;\mathsf{log1p}\left(\frac{y}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}\right) \cdot c\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3300:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -3300.0)
   (* (* (expm1 x) y) c)
   (*
    (log1p
     (*
      (*
       (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
       x)
      y))
    c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -3300.0) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -3300.0)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -3300.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3300

   1. Initial program 55.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 55.5

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

      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

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   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 61.3

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

   7. Applied rewrites 61.3%

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

   if -3300 < x

   1. Initial program 33.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 33.8

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

      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 34.9

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

      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 34.9

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

      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 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 x around 0

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

      1. *-commutative N/A

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

   7. Applied rewrites 90.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3300:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -250000:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -250000.0)
   (* (* (expm1 x) y) c)
   (* (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -250000.0) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p(((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -250000.0)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -250000.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e5

   1. Initial program 54.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 54.8

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

      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 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

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   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 62.1

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

   7. Applied rewrites 62.1%

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

   if -2.5e5 < x

   1. Initial program 34.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 34.1

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 35.2

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

      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 35.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 91.5

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

   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 x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 90.5

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

   7. Applied rewrites 90.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250000:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3300:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -3300.0)
   (* (* (expm1 x) y) c)
   (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -3300.0) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p(((fma(0.5, x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -3300.0)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(Float64(fma(0.5, x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -3300.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3300

   1. Initial program 55.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 55.5

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

      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

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   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 61.3

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

   7. Applied rewrites 61.3%

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

   if -3300 < x

   1. Initial program 33.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 33.8

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

      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 34.9

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

      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 34.9

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

      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 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 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. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3300:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -250000:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -250000.0) (* (* (expm1 x) y) c) (* (log1p (* x y)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -250000.0) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p((x * y)) * c;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -250000.0) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = Math.log1p((x * y)) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -250000.0:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = math.log1p((x * y)) * c
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -250000.0)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(x * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -250000.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e5

   1. Initial program 54.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 54.8

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

      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 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

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   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 62.1

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

   7. Applied rewrites 62.1%

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

   if -2.5e5 < x

   1. Initial program 34.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 34.1

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 35.2

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

      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 35.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 91.5

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

   4. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-expm1.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. pow2 N/A

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

      9. pow-exp N/A

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

      10. metadata-eval N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\color{blue}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

      12. lower-*.f64 91.5

          \[\leadsto \mathsf{log1p}\left(y \cdot \frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(\color{blue}{x \cdot 2}\right)}}\right) \cdot c \]

   6. Applied rewrites 91.5%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\frac{1}{\frac{e^{x} + 1}{\mathsf{expm1}\left(x \cdot 2\right)}}}\right) \cdot c \]

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 90.2

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

   9. Applied rewrites 90.2%

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

   10. Taylor expanded in x around 0

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

       1. lower-*.f64 89.7

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

   12. Applied rewrites 89.7%

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

Alternative 10: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-275}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot c, x, 0.16666666666666666 \cdot c\right), x, 0.5 \cdot c\right), x, c\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x 4.4e-275)
   (* (* (expm1 x) y) c)
   (*
    (*
     (fma
      (fma
       (fma (* 0.041666666666666664 c) x (* 0.16666666666666666 c))
       x
       (* 0.5 c))
      x
      c)
     x)
    y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= 4.4e-275) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (fma(fma(fma((0.041666666666666664 * c), x, (0.16666666666666666 * c)), x, (0.5 * c)), x, c) * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= 4.4e-275)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(fma(fma(fma(Float64(0.041666666666666664 * c), x, Float64(0.16666666666666666 * c)), x, Float64(0.5 * c)), x, c) * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, 4.4e-275], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(N[(N[(N[(0.041666666666666664 * c), $MachinePrecision] * x + N[(0.16666666666666666 * c), $MachinePrecision]), $MachinePrecision] * x + N[(0.5 * c), $MachinePrecision]), $MachinePrecision] * x + c), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.39999999999999977e-275

   1. Initial program 45.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.3

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 64.1

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

      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 64.1

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

      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 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      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 71.3

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

   7. Applied rewrites 71.3%

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

   if 4.39999999999999977e-275 < x

   1. Initial program 30.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-E.f64 30.2

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \left(c \cdot \log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{2} \cdot \left(c \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + x \cdot \left(\frac{1}{24} \cdot \left(c \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{4}\right)\right) + \frac{1}{6} \cdot \left(c \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right)\right)\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 73.3%

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

Alternative 11: 63.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 8.2 \cdot 10^{+27}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot c, x, c\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 8.2e+27)
   (* (* c y) x)
   (* (* (fma (* (fma 0.16666666666666666 x 0.5) c) x c) x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 8.2e+27) {
		tmp = (c * y) * x;
	} else {
		tmp = (fma((fma(0.16666666666666666, x, 0.5) * c), x, c) * x) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 8.2e+27)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(fma(Float64(fma(0.16666666666666666, x, 0.5) * c), x, c) * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 8.2e+27], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * c), $MachinePrecision] * x + c), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 8.2000000000000005e27

   1. Initial program 45.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-commutative N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. metadata-eval N/A

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

      17. log-E N/A

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

      18. log-E N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 8.2000000000000005e27 < c

   1. Initial program 21.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-E.f64 19.5

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

   5. Applied rewrites 19.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.2%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 60.2%

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

Alternative 12: 63.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 10^{+17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1e+17) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e+17) {
		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 <= 1d+17) 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 <= 1e+17) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1e+17:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1e+17)
		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 <= 1e+17)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1e+17], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1e17

   1. Initial program 46.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-commutative N/A

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

      4. *-lft-identity N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. metadata-eval N/A

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

      17. log-E N/A

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

      18. log-E N/A

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

      19. metadata-eval N/A

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

      20. *-rgt-identity N/A

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

      21. lower-*.f64 60.7

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

   5. Applied rewrites 60.7%

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

   if 1e17 < c

   1. Initial program 19.8%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-E.f64 19.7

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

   5. Applied rewrites 19.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.1%

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

Alternative 13: 61.6% 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 39.3%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. log-E N/A

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

   3. *-commutative N/A

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

   4. *-lft-identity N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-rgt-identity N/A

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

   9. metadata-eval N/A

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

   10. log-E N/A

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

   11. log-E N/A

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

   12. metadata-eval N/A

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

   13. log-E N/A

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

   14. lower-*.f64 N/A

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

   15. log-E N/A

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

   16. metadata-eval N/A

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

   17. log-E N/A

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

   18. log-E N/A

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

   19. metadata-eval N/A

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

   20. *-rgt-identity N/A

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

   21. lower-*.f64 57.9

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

5. Applied rewrites 57.9%

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

Developer Target 1: 93.6% 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 2024295 
(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)))))