Logarithmic Transform

Percentage Accurate: 42.0% → 99.1%

Time: 29.0s

Alternatives: 9

Speedup: 19.8×

• Could not converge on a ground truth

  1. c = 4.62310929775029e+56
  2. x = 2.0996510059109585e+200
  3. y = 7.041424521721224e-176

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: 42.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-92} \lor \neg \left(y \leq 1.2 \cdot 10^{-31}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -4.8e-92) (not (<= y 1.2e-31)))
   (* (log1p (* (expm1 x) y)) c)
   (* (* c y) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -4.8e-92) || !(y <= 1.2e-31)) {
		tmp = log1p((expm1(x) * y)) * c;
	} else {
		tmp = (c * y) * expm1(x);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -4.8e-92) || !(y <= 1.2e-31)) {
		tmp = Math.log1p((Math.expm1(x) * y)) * c;
	} else {
		tmp = (c * y) * Math.expm1(x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -4.8e-92) or not (y <= 1.2e-31):
		tmp = math.log1p((math.expm1(x) * y)) * c
	else:
		tmp = (c * y) * math.expm1(x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -4.8e-92) || !(y <= 1.2e-31))
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	else
		tmp = Float64(Float64(c * y) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -4.8e-92], N[Not[LessEqual[y, 1.2e-31]], $MachinePrecision]], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $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 < -4.8000000000000002e-92 or 1.2e-31 < y

   1. Initial program 36.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.0%

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

   if -4.8000000000000002e-92 < y < 1.2e-31

   1. Initial program 40.6%

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

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 99.8

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-92} \lor \neg \left(y \leq 1.2 \cdot 10^{-31}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\mathsf{E}\left(\right)}^{x} - 1 \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (- (pow (E) x) 1.0) -1e-12)
   (* (* (expm1 x) y) c)
   (* (log1p (* x y)) c)))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) < -9.9999999999999998e-13

   1. Initial program 41.8%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-expm1.f64 N/A

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

      2. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c \]

      3. lift-expm1.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-expm1.f64 N/A

         \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      7. lift-*.f64 71.0

         \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \color{blue}{y}\right) \cdot c \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      9. *-rgt-identity 71.0

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

   7. Applied rewrites 71.0%

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

   if -9.9999999999999998e-13 < (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64))

   1. Initial program 37.0%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 88.5%

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

Alternative 3: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -70:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -70.0)
   (* (log1p (* x y)) c)
   (if (<= y 2e+23)
     (* (* c y) (expm1 x))
     (*
      (log1p
       (*
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)
        y))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -70.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 2e+23) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -70.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 2e+23)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -70.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 2e+23], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -70

   1. Initial program 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.1%

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

   if -70 < y < 1.9999999999999998e23

   1. Initial program 38.0%

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

   3. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 99.4

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]

   if 1.9999999999999998e23 < y

   1. Initial program 23.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto c \cdot \log \left(1 + \left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + x \cdot \left(\frac{1}{24} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{4}\right) + \frac{1}{6} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2} + x \cdot \left(\frac{1}{24} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{4}\right) + \frac{1}{6} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \]

   5. Applied rewrites 48.4%

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, 1, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 48.4

         \[\leadsto \color{blue}{\log \left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, 1, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c} \]

   7. Applied rewrites 94.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -70:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -70:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -70.0)
   (* (log1p (* x y)) c)
   (if (<= y 2e+23)
     (* (* c y) (expm1 x))
     (* (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -70.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 2e+23) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = log1p(((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -70.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 2e+23)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -70.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 2e+23], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -70

   1. Initial program 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.1%

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

   if -70 < y < 1.9999999999999998e23

   1. Initial program 38.0%

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

   3. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 99.4

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]

   if 1.9999999999999998e23 < y

   1. Initial program 23.5%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. *-rgt-identity N/A

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

      6. metadata-eval N/A

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

      7. log-E N/A

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

      8. +-commutative N/A

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

      9. log-E N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 94.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -70:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -70:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -70.0)
   (* (log1p (* x y)) c)
   (if (<= y 2e+23)
     (* (* c y) (expm1 x))
     (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -70.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 2e+23) {
		tmp = (c * y) * expm1(x);
	} else {
		tmp = log1p(((fma(0.5, x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -70.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 2e+23)
		tmp = Float64(Float64(c * y) * expm1(x));
	else
		tmp = Float64(log1p(Float64(Float64(fma(0.5, x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -70.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 2e+23], N[(N[(c * y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -70

   1. Initial program 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.1%

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

   if -70 < y < 1.9999999999999998e23

   1. Initial program 38.0%

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

   3. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 99.4

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]

   if 1.9999999999999998e23 < y

   1. Initial program 23.5%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 94.6

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

   7. Applied rewrites 94.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -70:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -70 \lor \neg \left(y \leq 2 \cdot 10^{+23}\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -70.0) (not (<= y 2e+23)))
   (* (log1p (* x y)) c)
   (* (* c y) (expm1 x))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -70.0) || !(y <= 2e+23)) {
		tmp = log1p((x * y)) * c;
	} else {
		tmp = (c * y) * expm1(x);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -70.0) || !(y <= 2e+23)) {
		tmp = Math.log1p((x * y)) * c;
	} else {
		tmp = (c * y) * Math.expm1(x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -70.0) or not (y <= 2e+23):
		tmp = math.log1p((x * y)) * c
	else:
		tmp = (c * y) * math.expm1(x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -70.0) || !(y <= 2e+23))
		tmp = Float64(log1p(Float64(x * y)) * c);
	else
		tmp = Float64(Float64(c * y) * expm1(x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -70.0], N[Not[LessEqual[y, 2e+23]], $MachinePrecision]], N[(N[Log[1 + N[(x * y), $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 < -70 or 1.9999999999999998e23 < y

   1. Initial program 39.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.6%

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

   if -70 < y < 1.9999999999999998e23

   1. Initial program 38.0%

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

   3. 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

      8. lower-*.f64 99.4

         \[\leadsto \left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right) \]

   5. Applied rewrites 99.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -70 \lor \neg \left(y \leq 2 \cdot 10^{+23}\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-99}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -2.5e-99) (* (* (expm1 x) y) c) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -2.5e-99) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -2.5e-99) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -2.5e-99:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -2.5e-99)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -2.5e-99], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.49999999999999985e-99

   1. Initial program 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-expm1.f64 N/A

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

      2. *-rgt-identity N/A

         \[\leadsto \left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c \]

      3. lift-expm1.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-expm1.f64 N/A

         \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      7. lift-*.f64 74.8

         \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \color{blue}{y}\right) \cdot c \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right) \cdot c \]

      9. *-rgt-identity 74.8

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

   7. Applied rewrites 74.8%

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

   if -2.49999999999999985e-99 < x

   1. Initial program 36.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot 1\right) \]

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 82.8

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

   5. Applied rewrites 82.8%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 82.8

         \[\leadsto \left(c \cdot x\right) \cdot y \]

   7. Applied rewrites 82.8%

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

Alternative 8: 63.8% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 10^{-19}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1e-19) (* (* c y) x) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e-19) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * 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 <= 1d-19) then
        tmp = (c * y) * x
    else
        tmp = (c * x) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e-19) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1e-19:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1e-19)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1e-19)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1e-19], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 9.9999999999999998e-20

   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 x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot y\right) \cdot x \]
   7. Step-by-step derivation

      1. lift-*.f64 60.9

         \[\leadsto \left(c \cdot y\right) \cdot x \]

   8. Applied rewrites 60.9%

      \[\leadsto \left(c \cdot y\right) \cdot x \]

   if 9.9999999999999998e-20 < c

   1. Initial program 19.7%

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

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

      2. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot 1\right) \]

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 65.0

         \[\leadsto \left(c \cdot x\right) \cdot y \]

   7. Applied rewrites 65.0%

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

Alternative 9: 62.4% 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

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 38.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(c \cdot y\right) \cdot x \]
7. Step-by-step derivation

   1. lift-*.f64 59.4

      \[\leadsto \left(c \cdot y\right) \cdot x \]

8. Applied rewrites 59.4%

   \[\leadsto \left(c \cdot y\right) \cdot x \]
9. 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 2025037 
(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)))))