Logarithmic Transform

Percentage Accurate: 41.3% → 98.7%

Time: 26.6s

Alternatives: 11

Speedup: 19.8×

• Could not converge on a ground truth

  1. c = -4.809155554052532e-130
  2. x = 4.143761962601379e+24
  3. y = 2088.2826822129723

Specification

Math

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

FPCore

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

Initial Program: 41.3% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.00146 \lor \neg \left(y \leq 6 \cdot 10^{-119}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot c, {\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot y, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -0.00146) (not (<= y 6e-119)))
   (* (log1p (* (expm1 x) y)) c)
   (* (fma (* -0.5 c) (* (pow (expm1 x) 2.0) y) (* (expm1 x) c)) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -0.00146) || !(y <= 6e-119)) {
		tmp = log1p((expm1(x) * y)) * c;
	} else {
		tmp = fma((-0.5 * c), (pow(expm1(x), 2.0) * y), (expm1(x) * c)) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -0.00146) || !(y <= 6e-119))
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	else
		tmp = Float64(fma(Float64(-0.5 * c), Float64((expm1(x) ^ 2.0) * y), Float64(expm1(x) * c)) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -0.00146], N[Not[LessEqual[y, 6e-119]], $MachinePrecision]], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(-0.5 * c), $MachinePrecision] * N[(N[Power[N[(Exp[x] - 1), $MachinePrecision], 2.0], $MachinePrecision] * y), $MachinePrecision] + N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0014599999999999999 or 6.0000000000000004e-119 < y

   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 98.1%

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

   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 98.1%

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

   if -0.0014599999999999999 < y < 6.0000000000000004e-119

   1. Initial program 47.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot c, {\left(\mathsf{expm1}\left(x \cdot 1\right)\right)}^{2} \cdot y, \mathsf{expm1}\left(x \cdot 1\right) \cdot c\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.00146 \lor \neg \left(y \leq 6 \cdot 10^{-119}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot c, {\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot y, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-202} \lor \neg \left(y \leq 3.2 \cdot 10^{-274}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -2.2e-202) (not (<= y 3.2e-274)))
   (* (log1p (* (expm1 x) y)) c)
   (* (* c y) x)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -2.2e-202) || !(y <= 3.2e-274)) {
		tmp = log1p((expm1(x) * y)) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -2.2e-202) || !(y <= 3.2e-274)) {
		tmp = Math.log1p((Math.expm1(x) * y)) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -2.2e-202) or not (y <= 3.2e-274):
		tmp = math.log1p((math.expm1(x) * y)) * c
	else:
		tmp = (c * y) * x
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -2.2e-202) || !(y <= 3.2e-274))
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	else
		tmp = Float64(Float64(c * y) * x);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -2.2e-202], N[Not[LessEqual[y, 3.2e-274]], $MachinePrecision]], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000008e-202 or 3.19999999999999979e-274 < y

   1. Initial program 38.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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. 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 95.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 95.7%

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

   if -2.20000000000000008e-202 < y < 3.19999999999999979e-274

   1. Initial program 65.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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. 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 74.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lower-expm1.f64 N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. pow-to-exp N/A

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

      11. *-commutative N/A

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in x around 0

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

      1. lift-*.f64 97.2

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

   10. Applied rewrites 97.2%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-202} \lor \neg \left(y \leq 3.2 \cdot 10^{-274}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.028:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, y \cdot x, 0.16666666666666666 \cdot y\right), x, 0.5 \cdot y\right), x, y\right) \cdot x\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -0.028)
   (* (* (expm1 x) y) c)
   (*
    (log1p
     (*
      (fma
       (fma
        (fma 0.041666666666666664 (* y x) (* 0.16666666666666666 y))
        x
        (* 0.5 y))
       x
       y)
      x))
    c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -0.028) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p((fma(fma(fma(0.041666666666666664, (y * x), (0.16666666666666666 * y)), x, (0.5 * y)), x, y) * x)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -0.028)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(fma(fma(fma(0.041666666666666664, Float64(y * x), Float64(0.16666666666666666 * y)), x, Float64(0.5 * y)), x, y) * x)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -0.028], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(0.041666666666666664 * N[(y * x), $MachinePrecision] + N[(0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision] * x + N[(0.5 * y), $MachinePrecision]), $MachinePrecision] * x + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0280000000000000006

   1. Initial program 57.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 y around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 67.2

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 67.2

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

   7. Applied rewrites 67.2%

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

   if -0.0280000000000000006 < x

   1. Initial program 37.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 90.6%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 90.0%

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

Alternative 4: 84.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.028:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -0.028)
   (* (* (expm1 x) y) c)
   (*
    (log1p
     (*
      (*
       (fma (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5) x 1.0)
       x)
      y))
    c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -0.028) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -0.028)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -0.028], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0280000000000000006

   1. Initial program 57.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 y around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 67.2

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 67.2

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

   7. Applied rewrites 67.2%

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

   if -0.0280000000000000006 < x

   1. Initial program 37.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 90.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 90.0%

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

Alternative 5: 83.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.028:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -0.028)
   (* (* (expm1 x) y) c)
   (* (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -0.028) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p(((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -0.028)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -0.028], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0280000000000000006

   1. Initial program 57.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 y around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 67.2

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 67.2

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

   7. Applied rewrites 67.2%

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

   if -0.0280000000000000006 < x

   1. Initial program 37.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 90.6%

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

   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. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. log-E N/A

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

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. log-E N/A

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

      15. +-commutative N/A

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

      16. log-E N/A

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

   7. Applied rewrites 89.7%

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

Alternative 6: 74.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-274}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot x\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (* (expm1 x) y) c)))
   (if (<= y -2.2e-202)
     t_0
     (if (<= y 3.2e-274)
       (* (* c y) x)
       (if (<= y 5.9e+156) t_0 (* (log (* y x)) c))))))

C

double code(double c, double x, double y) {
	double t_0 = (expm1(x) * y) * c;
	double tmp;
	if (y <= -2.2e-202) {
		tmp = t_0;
	} else if (y <= 3.2e-274) {
		tmp = (c * y) * x;
	} else if (y <= 5.9e+156) {
		tmp = t_0;
	} else {
		tmp = log((y * x)) * c;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = (Math.expm1(x) * y) * c;
	double tmp;
	if (y <= -2.2e-202) {
		tmp = t_0;
	} else if (y <= 3.2e-274) {
		tmp = (c * y) * x;
	} else if (y <= 5.9e+156) {
		tmp = t_0;
	} else {
		tmp = Math.log((y * x)) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = (math.expm1(x) * y) * c
	tmp = 0
	if y <= -2.2e-202:
		tmp = t_0
	elif y <= 3.2e-274:
		tmp = (c * y) * x
	elif y <= 5.9e+156:
		tmp = t_0
	else:
		tmp = math.log((y * x)) * c
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(Float64(expm1(x) * y) * c)
	tmp = 0.0
	if (y <= -2.2e-202)
		tmp = t_0;
	elseif (y <= 3.2e-274)
		tmp = Float64(Float64(c * y) * x);
	elseif (y <= 5.9e+156)
		tmp = t_0;
	else
		tmp = Float64(log(Float64(y * x)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -2.2e-202], t$95$0, If[LessEqual[y, 3.2e-274], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 5.9e+156], t$95$0, N[(N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000008e-202 or 3.19999999999999979e-274 < y < 5.8999999999999997e156

   1. Initial program 42.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. 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 95.3%

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

   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. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 80.5

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 80.5

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

   7. Applied rewrites 80.5%

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

   if -2.20000000000000008e-202 < y < 3.19999999999999979e-274

   1. Initial program 65.9%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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. 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 74.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lower-expm1.f64 N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. pow-to-exp N/A

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

      11. *-commutative N/A

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

   7. Applied rewrites 89.0%

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

   8. Taylor expanded in x around 0

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

      1. lift-*.f64 97.2

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

   10. Applied rewrites 97.2%

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

   if 5.8999999999999997e156 < y

   1. Initial program 2.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. log-rec N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-log.f64 N/A

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

       9. lower-log.f64 78.4

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

   11. Applied rewrites 78.4%

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

   12. Taylor expanded in x around 0

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

       1. sum-log N/A

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

       2. lower-log.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 78.9

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

   14. Applied rewrites 78.9%

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

Alternative 7: 83.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.015:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -0.015)
   (* (* (expm1 x) y) c)
   (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -0.015) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p(((fma(0.5, x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -0.015)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(Float64(fma(0.5, x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -0.015], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.014999999999999999

   1. Initial program 57.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 y around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 67.2

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 67.2

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

   7. Applied rewrites 67.2%

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

   if -0.014999999999999999 < x

   1. Initial program 37.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 90.6%

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

   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. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. *-rgt-identity N/A

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

      11. lower-*.f64 N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 89.3

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

   7. Applied rewrites 89.3%

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

Alternative 8: 83.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.015:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -0.015) (* (* (expm1 x) y) c) (* (log1p (* x y)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -0.015) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = log1p((x * y)) * c;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -0.015) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = Math.log1p((x * y)) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -0.015:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = math.log1p((x * y)) * c
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -0.015)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(log1p(Float64(x * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -0.015], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.014999999999999999

   1. Initial program 57.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 y around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 67.2

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 67.2

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

   7. Applied rewrites 67.2%

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

   if -0.014999999999999999 < x

   1. Initial program 37.8%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 90.6%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 88.5

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

      4. *-rgt-identity 88.5

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

      5. *-commutative 88.5

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

      6. log-E 88.5

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

      7. pow-to-exp 88.5

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

   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 9: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+156}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot x\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y 4.8e+156) (* (* c y) x) (* (log (* y x)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 4.8e+156) {
		tmp = (c * y) * x;
	} else {
		tmp = log((y * x)) * c;
	}
	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 (y <= 4.8d+156) then
        tmp = (c * y) * x
    else
        tmp = log((y * x)) * c
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= 4.8e+156) {
		tmp = (c * y) * x;
	} else {
		tmp = Math.log((y * x)) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= 4.8e+156:
		tmp = (c * y) * x
	else:
		tmp = math.log((y * x)) * c
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 4.8e+156)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(log(Float64(y * x)) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (y <= 4.8e+156)
		tmp = (c * y) * x;
	else
		tmp = log((y * x)) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 4.8e+156], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.8000000000000002e156

   1. Initial program 45.6%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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. 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 92.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lower-expm1.f64 N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. pow-to-exp N/A

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

      11. *-commutative N/A

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

   7. Applied rewrites 68.2%

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

   8. Taylor expanded in x around 0

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

      1. lift-*.f64 72.8

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

   10. Applied rewrites 72.8%

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

   if 4.8000000000000002e156 < y

   1. Initial program 2.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. log-rec N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-log.f64 N/A

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

       9. lower-log.f64 78.4

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

   11. Applied rewrites 78.4%

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

   12. Taylor expanded in x around 0

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

       1. sum-log N/A

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

       2. lower-log.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 78.9

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

   14. Applied rewrites 78.9%

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

Alternative 10: 63.0% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 2.4 \cdot 10^{+179}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 2.4e+179) (* (* c y) x) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.4e+179) {
		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 <= 2.4d+179) 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 <= 2.4e+179) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 2.4e+179:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 2.4e+179)
		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 <= 2.4e+179)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 2.4e+179], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.40000000000000013e179

   1. Initial program 46.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   2. Add Preprocessing
   3. 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 93.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lower-expm1.f64 N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. pow-to-exp N/A

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

      11. *-commutative N/A

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

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in x around 0

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

      1. lift-*.f64 71.3

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

   10. Applied rewrites 71.3%

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

   if 2.40000000000000013e179 < c

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      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 59.0

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

   5. Applied rewrites 59.0%

      \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot 1\right)} \]
   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 59.0

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

   7. Applied rewrites 59.0%

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

Alternative 11: 61.8% 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 42.3%

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

   1. lift-*.f64 N/A

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

   2. 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 92.7%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. lift-expm1.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lower-expm1.f64 N/A

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

   6. *-commutative N/A

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

   7. *-rgt-identity N/A

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

   8. *-commutative N/A

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

   9. log-E N/A

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

   10. pow-to-exp N/A

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

   11. *-commutative N/A

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

7. Applied rewrites 62.9%

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

8. Taylor expanded in x around 0

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

   1. lift-*.f64 68.4

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

10. Applied rewrites 68.4%

    \[\leadsto \left(c \cdot y\right) \cdot x \]
11. Add Preprocessing

Developer Target 1: 93.9% 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 2025056 
(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)))))