Logarithmic Transform

Percentage Accurate: 42.6% → 98.7%

Time: 35.7s

Alternatives: 9

Speedup: 19.8×

• Could not converge on a ground truth

  1. c = -1.2125461201894213e-15
  2. x = 4.879550586871711e+160
  3. y = 2.9674112415550136e+182

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: 42.6% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -170000 \lor \neg \left(y \leq 9.2 \cdot 10^{-85}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -170000.0) (not (<= y 9.2e-85)))
   (* (log1p (* y (expm1 x))) c)
   (* (* y c) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -170000.0) || !(y <= 9.2e-85)) {
		tmp = log1p((y * expm1(x))) * c;
	} else {
		tmp = (y * c) * expm1(x);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -170000.0) || !(y <= 9.2e-85)) {
		tmp = Math.log1p((y * Math.expm1(x))) * c;
	} else {
		tmp = (y * c) * Math.expm1(x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -170000.0) or not (y <= 9.2e-85):
		tmp = math.log1p((y * math.expm1(x))) * c
	else:
		tmp = (y * c) * math.expm1(x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -170000.0) || !(y <= 9.2e-85))
		tmp = Float64(log1p(Float64(y * expm1(x))) * c);
	else
		tmp = Float64(Float64(y * c) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -170000.0], N[Not[LessEqual[y, 9.2e-85]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e5 or 9.2000000000000001e-85 < 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. 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 35.6

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

   4. Applied rewrites 99.7%

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

   if -1.7e5 < y < 9.2000000000000001e-85

   1. Initial program 46.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 99.9%

      \[\leadsto \left(c \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(1 \cdot x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -170000 \lor \neg \left(y \leq 9.2 \cdot 10^{-85}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+33} \lor \neg \left(y \leq 7.6 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -5e+33) (not (<= y 7.6e+41)))
   (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c)
   (* (* y c) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -5e+33) || !(y <= 7.6e+41)) {
		tmp = log1p((y * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c;
	} else {
		tmp = (y * c) * expm1(x);
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -5e+33) || !(y <= 7.6e+41))
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(y * c) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -5e+33], N[Not[LessEqual[y, 7.6e+41]], $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[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999973e33 or 7.6000000000000003e41 < y

   1. Initial program 29.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 29.4

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 83.3

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

   7. Applied rewrites 83.3%

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

   if -4.99999999999999973e33 < y < 7.6000000000000003e41

   1. Initial program 47.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)\right)} \]
   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 66.2

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

   5. Applied rewrites 66.2%

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

   7. Applied rewrites 96.3%

      \[\leadsto \left(c \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(1 \cdot x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+33} \lor \neg \left(y \leq 7.6 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+33} \lor \neg \left(y \leq 7.6 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -5e+33) (not (<= y 7.6e+41)))
   (* (log1p (* y (* (fma 0.5 x 1.0) x))) c)
   (* (* y c) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -5e+33) || !(y <= 7.6e+41)) {
		tmp = log1p((y * (fma(0.5, x, 1.0) * x))) * c;
	} else {
		tmp = (y * c) * expm1(x);
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -5e+33) || !(y <= 7.6e+41))
		tmp = Float64(log1p(Float64(y * Float64(fma(0.5, x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(y * c) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -5e+33], N[Not[LessEqual[y, 7.6e+41]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999973e33 or 7.6000000000000003e41 < y

   1. Initial program 29.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 29.4

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

   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 81.9

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

   7. Applied rewrites 81.9%

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

   if -4.99999999999999973e33 < y < 7.6000000000000003e41

   1. Initial program 47.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)\right)} \]
   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 66.2

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

   5. Applied rewrites 66.2%

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

   7. Applied rewrites 96.3%

      \[\leadsto \left(c \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(1 \cdot x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.3%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+33} \lor \neg \left(y \leq 7.6 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+55}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{+218}:\\ \;\;\;\;c \cdot \left(\left(\mathsf{fma}\left(x, \mathsf{fma}\left(-0.5 \cdot x - 0.5, y, \mathsf{fma}\left(0.16666666666666666, x, 0.5\right)\right), 1\right) \cdot y\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (fma y x 1.0)))))
   (if (<= y -2.5e+160)
     t_0
     (if (<= y 3.9e+55)
       (* (* y c) (expm1 x))
       (if (<= y 1.72e+218)
         (*
          c
          (*
           (*
            (fma
             x
             (fma (- (* -0.5 x) 0.5) y (fma 0.16666666666666666 x 0.5))
             1.0)
            y)
           x))
         t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = c * log(fma(y, x, 1.0));
	double tmp;
	if (y <= -2.5e+160) {
		tmp = t_0;
	} else if (y <= 3.9e+55) {
		tmp = (y * c) * expm1(x);
	} else if (y <= 1.72e+218) {
		tmp = c * ((fma(x, fma(((-0.5 * x) - 0.5), y, fma(0.16666666666666666, x, 0.5)), 1.0) * y) * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(fma(y, x, 1.0)))
	tmp = 0.0
	if (y <= -2.5e+160)
		tmp = t_0;
	elseif (y <= 3.9e+55)
		tmp = Float64(Float64(y * c) * expm1(x));
	elseif (y <= 1.72e+218)
		tmp = Float64(c * Float64(Float64(fma(x, fma(Float64(Float64(-0.5 * x) - 0.5), y, fma(0.16666666666666666, x, 0.5)), 1.0) * y) * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.5e+160], t$95$0, If[LessEqual[y, 3.9e+55], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.72e+218], N[(c * N[(N[(N[(x * N[(N[(N[(-0.5 * x), $MachinePrecision] - 0.5), $MachinePrecision] * y + N[(0.16666666666666666 * x + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5000000000000001e160 or 1.72e218 < 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. 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 68.1

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

   5. Applied rewrites 68.1%

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

   if -2.5000000000000001e160 < y < 3.90000000000000027e55

   1. Initial program 46.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 55.6

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

   5. Applied rewrites 55.6%

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

   7. Applied rewrites 88.9%

      \[\leadsto \left(c \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(1 \cdot x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 88.9%

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

   if 3.90000000000000027e55 < y < 1.72e218

   1. Initial program 24.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 32.5%

      \[\leadsto c \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(2, y, -3\right), y\right), \mathsf{fma}\left(y, y, -y\right) \cdot -0.5\right), x, y\right) \cdot x\right)} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.4%

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

Alternative 5: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 7 \cdot 10^{+132}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 7e+132) (* (* y c) (expm1 x)) (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 7e+132) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 7e+132) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 7e+132:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 7e+132)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 7e+132], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 7.00000000000000041e132

   1. Initial program 45.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 52.1

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

   5. Applied rewrites 52.1%

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

   7. Applied rewrites 79.9%

      \[\leadsto \left(c \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(1 \cdot x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 79.9%

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

   if 7.00000000000000041e132 < c

   1. Initial program 20.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 20.9

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 73.4

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

   7. Applied rewrites 73.4%

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

Alternative 6: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+45}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y 3e+45) (* (* y c) (expm1 x)) (* c (* y x))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 3e+45) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * (y * x);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= 3e+45) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = c * (y * x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= 3e+45:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = c * (y * x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 3e+45)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * Float64(y * x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 3e+45], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.00000000000000011e45

   1. Initial program 44.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 82.0%

      \[\leadsto \left(c \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(1 \cdot x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.0%

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

   if 3.00000000000000011e45 < y

   1. Initial program 16.4%

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

   3. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. associate-*r* N/A

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

      5. log-E N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

Alternative 7: 62.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 4.1 \cdot 10^{+32}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right) \cdot c, x, c\right) \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 4.1e+32)
   (* (* 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 <= 4.1e+32) {
		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 <= 4.1e+32)
		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, 4.1e+32], 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 < 4.09999999999999981e32

   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 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. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. log-E N/A

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

      11. lower-*.f64 N/A

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

      12. log-E N/A

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

      13. metadata-eval N/A

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

      14. log-E N/A

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

      15. log-E N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if 4.09999999999999981e32 < c

   1. Initial program 21.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.7%

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

   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 58.7%

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

Alternative 8: 62.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 7 \cdot 10^{+132}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 7e+132) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 7e+132) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 7d+132) 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 <= 7e+132) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 7e+132:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 7e+132)
		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 <= 7e+132)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 7e+132], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 7.00000000000000041e132

   1. Initial program 45.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   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. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. log-E N/A

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

      11. lower-*.f64 N/A

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

      12. log-E N/A

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

      13. metadata-eval N/A

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

      14. log-E N/A

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

      15. log-E N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 65.3

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

   5. Applied rewrites 65.3%

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

   if 7.00000000000000041e132 < c

   1. Initial program 20.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 17.6

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

   5. Applied rewrites 17.6%

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

   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 60.5%

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

Alternative 9: 60.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c, x, y)
use fmin_fmax_functions
    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 41.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. log-E N/A

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

   3. *-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. *-lft-identity N/A

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

   9. *-commutative N/A

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

   10. log-E N/A

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

   11. lower-*.f64 N/A

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

   12. log-E N/A

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

   13. metadata-eval N/A

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

   14. log-E N/A

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

   15. log-E N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity N/A

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

   18. lower-*.f64 63.0

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

5. Applied rewrites 63.0%

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

Developer Target 1: 93.3% 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 2024356 
(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)))))