Logarithmic Transform

Percentage Accurate: 41.6% → 98.8%

Time: 11.7s

Alternatives: 11

Speedup: 19.8×

• could not determine a ground truth

  1. c = 3.1197597569306404e+64
  2. x = 5.527766394054044e+294
  3. y = 1.3485790413558704e-174

Specification

Math

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

FPCore

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

Initial Program: 41.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-92} \lor \neg \left(y \leq 2.2 \cdot 10^{-51}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot y}{{\left(\mathsf{expm1}\left(x\right)\right)}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -4.7e-92) (not (<= y 2.2e-51)))
   (* (log1p (* y (expm1 x))) c)
   (/ (* c y) (pow (expm1 x) -1.0))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -4.7e-92) || !(y <= 2.2e-51)) {
		tmp = log1p((y * expm1(x))) * c;
	} else {
		tmp = (c * y) / pow(expm1(x), -1.0);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -4.7e-92) || !(y <= 2.2e-51)) {
		tmp = Math.log1p((y * Math.expm1(x))) * c;
	} else {
		tmp = (c * y) / Math.pow(Math.expm1(x), -1.0);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -4.7e-92) or not (y <= 2.2e-51):
		tmp = math.log1p((y * math.expm1(x))) * c
	else:
		tmp = (c * y) / math.pow(math.expm1(x), -1.0)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -4.7e-92) || !(y <= 2.2e-51))
		tmp = Float64(log1p(Float64(y * expm1(x))) * c);
	else
		tmp = Float64(Float64(c * y) / (expm1(x) ^ -1.0));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -4.7e-92], N[Not[LessEqual[y, 2.2e-51]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] / N[Power[N[(Exp[x] - 1), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.69999999999999993e-92 or 2.2e-51 < y

   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 42.0

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

      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 42.0

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

      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. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -4.69999999999999993e-92 < y < 2.2e-51

   1. Initial program 47.6%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-92} \lor \neg \left(y \leq 2.2 \cdot 10^{-51}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot y}{{\left(\mathsf{expm1}\left(x\right)\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\mathsf{fma}\left(\frac{x}{c}, -0.5, {c}^{-1}\right)}{x}\right)}^{-1} \cdot y \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (* (pow (/ (fma (/ x c) -0.5 (pow c -1.0)) x) -1.0) y))

C

double code(double c, double x, double y) {
	return pow((fma((x / c), -0.5, pow(c, -1.0)) / x), -1.0) * y;
}

Julia

function code(c, x, y)
	return Float64((Float64(fma(Float64(x / c), -0.5, (c ^ -1.0)) / x) ^ -1.0) * y)
end

Wolfram

code[c_, x_, y_] := N[(N[Power[N[(N[(N[(x / c), $MachinePrecision] * -0.5 + N[Power[c, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 40.5%

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

3. Taylor expanded in 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 45.1

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

5. Applied rewrites 45.1%

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

7. Applied rewrites 76.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\frac{\frac{-1}{2} \cdot \frac{x}{c} + \frac{1}{c}}{x}} \cdot y \]
9. Step-by-step derivation

10. Applied rewrites 63.7%

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

11. Final simplification 63.7%

    \[\leadsto {\left(\frac{\mathsf{fma}\left(\frac{x}{c}, -0.5, {c}^{-1}\right)}{x}\right)}^{-1} \cdot y \]
12. Add Preprocessing

Alternative 3: 76.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+20}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot c}{\mathsf{fma}\left(-0.5 \cdot x, \frac{y}{y \cdot y} - 1, {y}^{-1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y 3e+20)
   (* (* c y) (expm1 x))
   (/ (* x c) (fma (* -0.5 x) (- (/ y (* y y)) 1.0) (pow y -1.0)))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 3e+20) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = (x * c) / fma((-0.5 * x), ((y / (y * y)) - 1.0), pow(y, -1.0));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 3e+20)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(Float64(x * c) / fma(Float64(-0.5 * x), Float64(Float64(y / Float64(y * y)) - 1.0), (y ^ -1.0)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 3e+20], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(x * c), $MachinePrecision] / N[(N[(-0.5 * x), $MachinePrecision] * N[(N[(y / N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] + N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e20

   1. Initial program 45.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 51.5

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

   5. Applied rewrites 51.5%

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

   7. Applied rewrites 80.4%

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

   if 3e20 < y

   1. Initial program 12.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 42.7%

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

   7. Applied rewrites 47.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.5%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+20}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot c}{\mathsf{fma}\left(-0.5 \cdot x, \frac{y}{y \cdot y} - 1, {y}^{-1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -5.2e+64)
   (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c)
   (if (<= y 7.2e-17)
     (* (* c y) (expm1 x))
     (*
      (log1p
       (*
        y
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -5.2e+64) {
		tmp = log1p((y * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c;
	} else if (y <= 7.2e-17) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = log1p((y * (fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x))) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -5.2e+64)
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c);
	elseif (y <= 7.2e-17)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x))) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -5.2e+64], N[(N[Log[1 + N[(y * N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 7.2e-17], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(y * N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.19999999999999994e64

   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 45.3

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

      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 45.3

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{{1}^{3}}\right) + 1\right)\right)\right) \cdot c \]

      7. log-E N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \left(\frac{1}{6} \cdot x\right) \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{3}\right) + 1\right)\right)\right) \cdot c \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \color{blue}{\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}\right) + 1\right)\right)\right) \cdot c \]

      9. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)} + 1\right)\right)\right) \cdot c \]

      10. +-commutative N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)}\right)\right) \cdot c \]

      11. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(\color{blue}{\log \mathsf{E}\left(\right)} + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)\right) \cdot c \]

      12. *-commutative N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot x\right)}\right) \cdot c \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot x\right)}\right) \cdot c \]

   7. Applied rewrites 65.0%

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

   if -5.19999999999999994e64 < y < 7.1999999999999999e-17

   1. Initial program 45.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 96.2%

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

   if 7.1999999999999999e-17 < y

   1. Initial program 19.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 19.1

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

      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 20.1

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

      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 20.1

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, 1\right)}{c}}{x}\right)}^{-1} \cdot y \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (* (pow (/ (/ (fma (fma 0.08333333333333333 x -0.5) x 1.0) c) x) -1.0) y))

C

double code(double c, double x, double y) {
	return pow(((fma(fma(0.08333333333333333, x, -0.5), x, 1.0) / c) / x), -1.0) * y;
}

Julia

function code(c, x, y)
	return Float64((Float64(Float64(fma(fma(0.08333333333333333, x, -0.5), x, 1.0) / c) / x) ^ -1.0) * y)
end

Wolfram

code[c_, x_, y_] := N[(N[Power[N[(N[(N[(N[(0.08333333333333333 * x + -0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] / c), $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 40.5%

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

3. Taylor expanded in 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 45.1

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

5. Applied rewrites 45.1%

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

7. Applied rewrites 76.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 62.3%

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

11. Taylor expanded in c around 0

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

13. Applied rewrites 62.3%

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

14. Final simplification 62.3%

    \[\leadsto {\left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x, -0.5\right), x, 1\right)}{c}}{x}\right)}^{-1} \cdot y \]
15. Add Preprocessing

Alternative 6: 88.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+64} \lor \neg \left(y \leq 7.2 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -5.2e+64) (not (<= y 7.2e-17)))
   (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c)
   (* (* c y) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -5.2e+64) || !(y <= 7.2e-17)) {
		tmp = log1p((y * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c;
	} else {
		tmp = (c * y) * expm1(x);
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -5.2e+64) || !(y <= 7.2e-17))
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(c * y) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -5.2e+64], N[Not[LessEqual[y, 7.2e-17]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999994e64 or 7.1999999999999999e-17 < y

   1. Initial program 32.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 32.1

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

      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 32.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 32.6

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{{1}^{3}}\right) + 1\right)\right)\right) \cdot c \]

      7. log-E N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \left(\frac{1}{6} \cdot x\right) \cdot {\color{blue}{\log \mathsf{E}\left(\right)}}^{3}\right) + 1\right)\right)\right) \cdot c \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + \color{blue}{\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}\right) + 1\right)\right)\right) \cdot c \]

      9. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)} + 1\right)\right)\right) \cdot c \]

      10. +-commutative N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)}\right)\right) \cdot c \]

      11. log-E N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \left(x \cdot \left(\color{blue}{\log \mathsf{E}\left(\right)} + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)\right) \cdot c \]

      12. *-commutative N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot x\right)}\right) \cdot c \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot x\right)}\right) \cdot c \]

   7. Applied rewrites 82.0%

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

   if -5.19999999999999994e64 < y < 7.1999999999999999e-17

   1. Initial program 45.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 96.2%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+64} \lor \neg \left(y \leq 7.2 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+64} \lor \neg \left(y \leq 7.2 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -5.2e+64) (not (<= y 7.2e-17)))
   (* (log1p (* y (* (fma 0.5 x 1.0) x))) c)
   (* (* c y) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -5.2e+64) || !(y <= 7.2e-17)) {
		tmp = log1p((y * (fma(0.5, x, 1.0) * x))) * c;
	} else {
		tmp = (c * y) * expm1(x);
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -5.2e+64) || !(y <= 7.2e-17))
		tmp = Float64(log1p(Float64(y * Float64(fma(0.5, x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(c * y) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -5.2e+64], N[Not[LessEqual[y, 7.2e-17]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999994e64 or 7.1999999999999999e-17 < y

   1. Initial program 32.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 32.1

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

      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 32.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 32.6

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if -5.19999999999999994e64 < y < 7.1999999999999999e-17

   1. Initial program 45.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

   7. Applied rewrites 96.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+64} \lor \neg \left(y \leq 7.2 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+117} \lor \neg \left(y \leq 5.2 \cdot 10^{+132}\right):\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -8.6e+117) (not (<= y 5.2e+132)))
   (* c (log (fma y x 1.0)))
   (* (* c y) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -8.6e+117) || !(y <= 5.2e+132)) {
		tmp = c * log(fma(y, x, 1.0));
	} else {
		tmp = (c * y) * expm1(x);
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -8.6e+117) || !(y <= 5.2e+132))
		tmp = Float64(c * log(fma(y, x, 1.0)));
	else
		tmp = Float64(Float64(c * y) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -8.6e+117], N[Not[LessEqual[y, 5.2e+132]], $MachinePrecision]], N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.59999999999999996e117 or 5.2e132 < y

   1. Initial program 35.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if -8.59999999999999996e117 < y < 5.2e132

   1. Initial program 42.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

   5. Applied rewrites 56.4%

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

   7. Applied rewrites 93.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+117} \lor \neg \left(y \leq 5.2 \cdot 10^{+132}\right):\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 2.5 \cdot 10^{+42}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 2.5e+42)
   (* (* c y) x)
   (* (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.5e+42) {
		tmp = (c * y) * x;
	} else {
		tmp = ((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 2.5e+42)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 2.5e+42], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.50000000000000003e42

   1. Initial program 46.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-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. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative 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. *-rgt-identity N/A

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

      17. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if 2.50000000000000003e42 < c

   1. Initial program 17.5%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 20.3

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.7%

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

Alternative 10: 63.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 0.1:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 0.1) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 0.1) {
		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 <= 0.1d0) 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 <= 0.1) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 0.1:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 0.1)
		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 <= 0.1)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 0.1], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 0.10000000000000001

   1. Initial program 47.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-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. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative 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. *-rgt-identity N/A

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

      17. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   if 0.10000000000000001 < c

   1. Initial program 17.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 20.3

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

   5. Applied rewrites 20.3%

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

   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 50.6%

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

Alternative 11: 61.4% 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 40.5%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. log-E N/A

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

   3. *-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. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative 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. *-rgt-identity N/A

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

   17. lower-*.f64 61.3

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

5. Applied rewrites 61.3%

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

Developer Target 1: 93.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024322 
(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)))))