Logarithmic Transform

Percentage Accurate: 41.5% → 98.4%

Time: 34.8s

Alternatives: 9

Speedup: 19.8×

• Could not converge on a ground truth

  1. c = 1.3304207953922887e+119
  2. x = 1.4101525860103042e+220
  3. y = 1.3179088232921843e+198

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: 41.5% 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: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-12} \lor \neg \left(y \leq 5 \cdot 10^{-144}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(y, -0.5 \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}, \mathsf{expm1}\left(x\right)\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -2e-12) (not (<= y 5e-144)))
   (* (log1p (* y (expm1 x))) c)
   (* (* (fma y (* -0.5 (pow (expm1 x) 2.0)) (expm1 x)) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -2e-12) || !(y <= 5e-144)) {
		tmp = log1p((y * expm1(x))) * c;
	} else {
		tmp = (fma(y, (-0.5 * pow(expm1(x), 2.0)), expm1(x)) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -2e-12) || !(y <= 5e-144))
		tmp = Float64(log1p(Float64(y * expm1(x))) * c);
	else
		tmp = Float64(Float64(fma(y, Float64(-0.5 * (expm1(x) ^ 2.0)), expm1(x)) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -2e-12], N[Not[LessEqual[y, 5e-144]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(y * N[(-0.5 * N[Power[N[(Exp[x] - 1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999996e-12 or 4.9999999999999998e-144 < y

   1. Initial program 35.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 99.7%

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

   if -1.99999999999999996e-12 < y < 4.9999999999999998e-144

   1. Initial program 61.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 61.4

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{2} \cdot -0.5\right) \cdot y, c, \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-12} \lor \neg \left(y \leq 5 \cdot 10^{-144}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(y, -0.5 \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}, \mathsf{expm1}\left(x\right)\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-17} \lor \neg \left(y \leq 5 \cdot 10^{-144}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -3.8e-17) (not (<= y 5e-144)))
   (* (log1p (* y (expm1 x))) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -3.8e-17) || !(y <= 5e-144)) {
		tmp = log1p((y * expm1(x))) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -3.8e-17) || !(y <= 5e-144)) {
		tmp = Math.log1p((y * Math.expm1(x))) * c;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -3.8e-17) or not (y <= 5e-144):
		tmp = math.log1p((y * math.expm1(x))) * c
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -3.8e-17) || !(y <= 5e-144))
		tmp = Float64(log1p(Float64(y * expm1(x))) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -3.8e-17], N[Not[LessEqual[y, 5e-144]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8000000000000001e-17 or 4.9999999999999998e-144 < y

   1. Initial program 35.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 35.8

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

   4. Applied rewrites 99.7%

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

   if -3.8000000000000001e-17 < y < 4.9999999999999998e-144

   1. Initial program 61.7%

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

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

      3. lower-*.f64 61.7

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-17} \lor \neg \left(y \leq 5 \cdot 10^{-144}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+19} \lor \neg \left(y \leq 1.1 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -4.2e+19) (not (<= y 1.1e-12)))
   (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -4.2e+19) || !(y <= 1.1e-12)) {
		tmp = log1p((y * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -4.2e+19) || !(y <= 1.1e-12))
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -4.2e+19], N[Not[LessEqual[y, 1.1e-12]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2e19 or 1.09999999999999996e-12 < y

   1. Initial program 33.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 33.4

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 82.8

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

   7. Applied rewrites 82.8%

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

   if -4.2e19 < y < 1.09999999999999996e-12

   1. Initial program 54.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 54.8

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 99.0

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

   7. Applied rewrites 99.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+19} \lor \neg \left(y \leq 1.1 \cdot 10^{-12}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(\mathsf{fma}\left(\left(-0.5 \cdot y\right) \cdot x, x, \mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right)\\ t_1 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{+48}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+186}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (* (fma (* (* -0.5 y) x) x (* (fma 0.5 x 1.0) x)) y)))
        (t_1 (* c (log (fma y x 1.0)))))
   (if (<= y -1.05e+217)
     t_1
     (if (<= y -4.8e+61)
       t_0
       (if (<= y 1e+48)
         (* (* (expm1 x) c) y)
         (if (<= y 1.26e+186) t_0 t_1))))))

C

double code(double c, double x, double y) {
	double t_0 = c * (fma(((-0.5 * y) * x), x, (fma(0.5, x, 1.0) * x)) * y);
	double t_1 = c * log(fma(y, x, 1.0));
	double tmp;
	if (y <= -1.05e+217) {
		tmp = t_1;
	} else if (y <= -4.8e+61) {
		tmp = t_0;
	} else if (y <= 1e+48) {
		tmp = (expm1(x) * c) * y;
	} else if (y <= 1.26e+186) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * Float64(fma(Float64(Float64(-0.5 * y) * x), x, Float64(fma(0.5, x, 1.0) * x)) * y))
	t_1 = Float64(c * log(fma(y, x, 1.0)))
	tmp = 0.0
	if (y <= -1.05e+217)
		tmp = t_1;
	elseif (y <= -4.8e+61)
		tmp = t_0;
	elseif (y <= 1e+48)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (y <= 1.26e+186)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[(N[(N[(N[(-0.5 * y), $MachinePrecision] * x), $MachinePrecision] * x + N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+217], t$95$1, If[LessEqual[y, -4.8e+61], t$95$0, If[LessEqual[y, 1e+48], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.26e+186], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e217 or 1.26e186 < y

   1. Initial program 35.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -1.05e217 < y < -4.7999999999999998e61 or 1.00000000000000004e48 < y < 1.26e186

   1. Initial program 34.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. log-E N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 10.4%

      \[\leadsto c \cdot \left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot -0.5\right) \cdot x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.2%

       \[\leadsto c \cdot \left(\left(\left(\left(x \cdot y\right) \cdot y\right) \cdot -0.5\right) \cdot x\right) \]

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 67.7%

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

   if -4.7999999999999998e61 < y < 1.00000000000000004e48

   1. Initial program 50.7%

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

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

      3. lower-*.f64 50.7

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 95.5

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

   7. Applied rewrites 95.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+217}:\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{+61}:\\ \;\;\;\;c \cdot \left(\mathsf{fma}\left(\left(-0.5 \cdot y\right) \cdot x, x, \mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right)\\ \mathbf{elif}\;y \leq 10^{+48}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+186}:\\ \;\;\;\;c \cdot \left(\mathsf{fma}\left(\left(-0.5 \cdot y\right) \cdot x, x, \mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+57} \lor \neg \left(y \leq 4.5 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -2.7e+57) (not (<= y 4.5e-7)))
   (* (log1p (* y (* (fma 0.5 x 1.0) x))) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -2.7e+57) || !(y <= 4.5e-7)) {
		tmp = log1p((y * (fma(0.5, x, 1.0) * x))) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -2.7e+57) || !(y <= 4.5e-7))
		tmp = Float64(log1p(Float64(y * Float64(fma(0.5, x, 1.0) * x))) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -2.7e+57], N[Not[LessEqual[y, 4.5e-7]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6999999999999998e57 or 4.4999999999999998e-7 < y

   1. Initial program 32.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.1

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 82.2

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

   7. Applied rewrites 82.2%

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

   if -2.6999999999999998e57 < y < 4.4999999999999998e-7

   1. Initial program 53.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 53.8

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 97.3

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

   7. Applied rewrites 97.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+57} \lor \neg \left(y \leq 4.5 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.46 \cdot 10^{+69} \lor \neg \left(y \leq 2.45 \cdot 10^{+46}\right):\\ \;\;\;\;c \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1.46e+69) (not (<= y 2.45e+46)))
   (* c (* y x))
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.46e+69) || !(y <= 2.45e+46)) {
		tmp = c * (y * x);
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.46e+69) || !(y <= 2.45e+46)) {
		tmp = c * (y * x);
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -1.46e+69) or not (y <= 2.45e+46):
		tmp = c * (y * x)
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1.46e+69) || !(y <= 2.45e+46))
		tmp = Float64(c * Float64(y * x));
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1.46e+69], N[Not[LessEqual[y, 2.45e+46]], $MachinePrecision]], N[(c * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.46000000000000007e69 or 2.44999999999999984e46 < y

   1. Initial program 36.2%

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

   3. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

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

      4. associate-*r* N/A

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

      5. log-E N/A

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

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 47.3

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

   5. Applied rewrites 47.3%

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

   if -1.46000000000000007e69 < y < 2.44999999999999984e46

   1. Initial program 49.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 49.9

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

   4. Applied rewrites 95.2%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 95.8

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

   7. Applied rewrites 95.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.46 \cdot 10^{+69} \lor \neg \left(y \leq 2.45 \cdot 10^{+46}\right):\\ \;\;\;\;c \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 5.1 \cdot 10^{-18}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 5.1e-18)
   (* (* c y) x)
   (* (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 5.1e-18) {
		tmp = (c * y) * x;
	} else {
		tmp = ((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * c) * y;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 5.1e-18)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 5.1e-18], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.09999999999999983e-18

   1. Initial program 52.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. log-E N/A

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

      11. lower-*.f64 N/A

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

      12. log-E N/A

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

      13. metadata-eval N/A

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

      14. log-E N/A

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

      15. log-E N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 5.09999999999999983e-18 < c

   1. Initial program 26.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 26.2

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-expm1.f64 75.5

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 53.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 5.1 \cdot 10^{-18}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{+70}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 4e+70) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 4e+70) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 4d+70) then
        tmp = (c * y) * x
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 4e+70) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 4e+70:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 4e+70)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 4e+70)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 4e+70], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 4.00000000000000029e70

   1. Initial program 51.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. log-E N/A

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

      11. lower-*.f64 N/A

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

      12. log-E N/A

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

      13. metadata-eval N/A

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

      14. log-E N/A

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

      15. log-E N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if 4.00000000000000029e70 < c

   1. Initial program 20.0%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. log-E N/A

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

      11. lower-*.f64 N/A

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

      12. log-E N/A

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

      13. metadata-eval N/A

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

      14. log-E N/A

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

      15. log-E N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. lower-*.f64 39.8

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

   5. Applied rewrites 39.8%

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

   7. Applied rewrites 48.4%

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

Alternative 9: 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 45.9%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. log-E N/A

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

   3. *-commutative N/A

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

   4. *-lft-identity N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-lft-identity N/A

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

   9. *-commutative N/A

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

   10. log-E N/A

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

   11. lower-*.f64 N/A

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

   12. log-E N/A

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

   13. metadata-eval N/A

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

   14. log-E N/A

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

   15. log-E N/A

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

   16. metadata-eval N/A

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

   17. *-rgt-identity N/A

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

   18. lower-*.f64 61.9

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

5. Applied rewrites 61.9%

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

Developer Target 1: 94.0% 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 2024358 
(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)))))