Logarithmic Transform

Percentage Accurate: 41.6% → 93.1%

Time: 12.7s

Alternatives: 8

Speedup: 19.8×

• could not determine a ground truth

  1. c = -4.049183138964582e-11
  2. x = 2.4848407585638556e+89
  3. y = 2.7447660737196446e-278

Specification

Math

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

FPCore

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

Initial Program: 41.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-208}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= y -1.8e-274) t_0 (if (<= y 1.45e-208) (* (* c y) x) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -1.8e-274) {
		tmp = t_0;
	} else if (y <= 1.45e-208) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((Math.expm1(x) * y));
	double tmp;
	if (y <= -1.8e-274) {
		tmp = t_0;
	} else if (y <= 1.45e-208) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((math.expm1(x) * y))
	tmp = 0
	if y <= -1.8e-274:
		tmp = t_0
	elif y <= 1.45e-208:
		tmp = (c * y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(expm1(x) * y)))
	tmp = 0.0
	if (y <= -1.8e-274)
		tmp = t_0;
	elseif (y <= 1.45e-208)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e-274], t$95$0, If[LessEqual[y, 1.45e-208], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999991e-274 or 1.45e-208 < y

   1. Initial program 33.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 33.7

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 50.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 50.1

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 96.2

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

   4. Applied rewrites 96.2%

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

   if -1.79999999999999991e-274 < y < 1.45e-208

   1. Initial program 49.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. *-rgt-identity N/A

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

      17. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-274}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-208}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-10}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -2.6e-10) (* (* (expm1 x) y) c) (* (log1p (* x y)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -2.6e-10) {
		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 <= -2.6e-10) {
		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 <= -2.6e-10:
		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 <= -2.6e-10)
		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, -2.6e-10], 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 < -2.59999999999999981e-10

   1. Initial program 47.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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. *-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 47.0

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 98.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.4

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 72.5

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

   7. Applied rewrites 72.5%

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

   if -2.59999999999999981e-10 < x

   1. Initial program 30.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 30.7

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 31.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.8

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 3: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-29}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -3.8e-29) (* (* (expm1 x) y) c) (* (* c y) x)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -3.8e-29) {
		tmp = (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 (x <= -3.8e-29) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -3.8e-29:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = (c * y) * x
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -3.8e-29)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(c * y) * x);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -3.8e-29], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.79999999999999976e-29

   1. Initial program 43.7%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 43.7

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 91.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 91.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lift-E.f64 N/A

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

      14. log-E N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 71.9

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

   7. Applied rewrites 71.9%

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

   if -3.79999999999999976e-29 < x

   1. Initial program 31.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. *-rgt-identity N/A

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

      17. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

Alternative 4: 62.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -66:\\ \;\;\;\;\left(\left(\left(-y\right) \cdot y\right) \cdot \frac{x}{\left(0.5 \cdot \left(y - y \cdot y\right)\right) \cdot x - y}\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -66.0)
   (* (* (* (- y) y) (/ x (- (* (* 0.5 (- y (* y y))) x) y))) c)
   (* (* c y) x)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -66.0) {
		tmp = ((-y * y) * (x / (((0.5 * (y - (y * y))) * x) - y))) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(c, x, y)
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-66.0d0)) then
        tmp = ((-y * y) * (x / (((0.5d0 * (y - (y * y))) * x) - y))) * c
    else
        tmp = (c * y) * x
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -66.0) {
		tmp = ((-y * y) * (x / (((0.5 * (y - (y * y))) * x) - y))) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -66.0:
		tmp = ((-y * y) * (x / (((0.5 * (y - (y * y))) * x) - y))) * c
	else:
		tmp = (c * y) * x
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -66.0)
		tmp = Float64(Float64(Float64(Float64(-y) * y) * Float64(x / Float64(Float64(Float64(0.5 * Float64(y - Float64(y * y))) * x) - y))) * c);
	else
		tmp = Float64(Float64(c * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (x <= -66.0)
		tmp = ((-y * y) * (x / (((0.5 * (y - (y * y))) * x) - y))) * c;
	else
		tmp = (c * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -66.0], N[(N[(N[((-y) * y), $MachinePrecision] * N[(x / N[(N[(N[(0.5 * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -66

   1. Initial program 49.4%

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

   3. Taylor expanded in x around 0

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

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 4.4%

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

   7. Applied rewrites 4.2%

      \[\leadsto c \cdot \frac{\mathsf{fma}\left({\left(\left(y - y \cdot y\right) \cdot x\right)}^{2}, 0.25, \left(-y\right) \cdot y\right) \cdot x}{\color{blue}{\left(\left(y - y \cdot y\right) \cdot 0.5\right) \cdot x - y}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 25.0%

       \[\leadsto c \cdot \frac{\left(\left(-y\right) \cdot y\right) \cdot x}{\color{blue}{\left(\left(y - y \cdot y\right) \cdot 0.5\right)} \cdot x - y} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 25.0

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

   12. Applied rewrites 25.2%

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

   if -66 < x

   1. Initial program 30.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. *-rgt-identity N/A

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

      17. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -66:\\ \;\;\;\;\left(\left(\left(-y\right) \cdot y\right) \cdot \frac{x}{\left(0.5 \cdot \left(y - y \cdot y\right)\right) \cdot x - y}\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -66:\\ \;\;\;\;\frac{\left(\left(-y\right) \cdot y\right) \cdot x}{\left(0.5 \cdot \left(y - y \cdot y\right)\right) \cdot x - y} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -66.0)
   (* (/ (* (* (- y) y) x) (- (* (* 0.5 (- y (* y y))) x) y)) c)
   (* (* c y) x)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -66.0) {
		tmp = (((-y * y) * x) / (((0.5 * (y - (y * y))) * x) - y)) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(c, x, y)
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-66.0d0)) then
        tmp = (((-y * y) * x) / (((0.5d0 * (y - (y * y))) * x) - y)) * c
    else
        tmp = (c * y) * x
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -66.0) {
		tmp = (((-y * y) * x) / (((0.5 * (y - (y * y))) * x) - y)) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -66.0:
		tmp = (((-y * y) * x) / (((0.5 * (y - (y * y))) * x) - y)) * c
	else:
		tmp = (c * y) * x
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -66.0)
		tmp = Float64(Float64(Float64(Float64(Float64(-y) * y) * x) / Float64(Float64(Float64(0.5 * Float64(y - Float64(y * y))) * x) - y)) * c);
	else
		tmp = Float64(Float64(c * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (x <= -66.0)
		tmp = (((-y * y) * x) / (((0.5 * (y - (y * y))) * x) - y)) * c;
	else
		tmp = (c * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -66.0], N[(N[(N[(N[((-y) * y), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[(0.5 * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -66

   1. Initial program 49.4%

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

   3. Taylor expanded in x around 0

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

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 4.4%

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

   7. Applied rewrites 4.2%

      \[\leadsto c \cdot \frac{\mathsf{fma}\left({\left(\left(y - y \cdot y\right) \cdot x\right)}^{2}, 0.25, \left(-y\right) \cdot y\right) \cdot x}{\color{blue}{\left(\left(y - y \cdot y\right) \cdot 0.5\right) \cdot x - y}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 25.0%

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

   if -66 < x

   1. Initial program 30.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. *-rgt-identity N/A

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

      17. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -66:\\ \;\;\;\;\frac{\left(\left(-y\right) \cdot y\right) \cdot x}{\left(0.5 \cdot \left(y - y \cdot y\right)\right) \cdot x - y} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -66:\\ \;\;\;\;\frac{\left(\left(-y\right) \cdot y\right) \cdot x}{\left(\mathsf{fma}\left(-y, y, y\right) \cdot x\right) \cdot 0.5} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -66.0)
   (* (/ (* (* (- y) y) x) (* (* (fma (- y) y y) x) 0.5)) c)
   (* (* c y) x)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -66.0) {
		tmp = (((-y * y) * x) / ((fma(-y, y, y) * x) * 0.5)) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -66.0)
		tmp = Float64(Float64(Float64(Float64(Float64(-y) * y) * x) / Float64(Float64(fma(Float64(-y), y, y) * x) * 0.5)) * c);
	else
		tmp = Float64(Float64(c * y) * x);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -66.0], N[(N[(N[(N[((-y) * y), $MachinePrecision] * x), $MachinePrecision] / N[(N[(N[((-y) * y + y), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -66

   1. Initial program 49.4%

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

   3. Taylor expanded in x around 0

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

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 4.4%

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

   7. Applied rewrites 4.2%

      \[\leadsto c \cdot \frac{\mathsf{fma}\left({\left(\left(y - y \cdot y\right) \cdot x\right)}^{2}, 0.25, \left(-y\right) \cdot y\right) \cdot x}{\color{blue}{\left(\left(y - y \cdot y\right) \cdot 0.5\right) \cdot x - y}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 25.0%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 25.0%

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

   if -66 < x

   1. Initial program 30.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. *-rgt-identity N/A

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

      17. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -66:\\ \;\;\;\;\frac{\left(\left(-y\right) \cdot y\right) \cdot x}{\left(\mathsf{fma}\left(-y, y, y\right) \cdot x\right) \cdot 0.5} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -110:\\ \;\;\;\;\frac{\left(\left(-y\right) \cdot y\right) \cdot x}{\mathsf{fma}\left(0.5, x, -1\right) \cdot y} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -110.0)
   (* (/ (* (* (- y) y) x) (* (fma 0.5 x -1.0) y)) c)
   (* (* c y) x)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -110.0) {
		tmp = (((-y * y) * x) / (fma(0.5, x, -1.0) * y)) * c;
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -110.0)
		tmp = Float64(Float64(Float64(Float64(Float64(-y) * y) * x) / Float64(fma(0.5, x, -1.0) * y)) * c);
	else
		tmp = Float64(Float64(c * y) * x);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -110.0], N[(N[(N[(N[((-y) * y), $MachinePrecision] * x), $MachinePrecision] / N[(N[(0.5 * x + -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -110

   1. Initial program 49.4%

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

   3. Taylor expanded in x around 0

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

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

   5. Applied rewrites 4.4%

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

   7. Applied rewrites 4.2%

      \[\leadsto c \cdot \frac{\mathsf{fma}\left({\left(\left(y - y \cdot y\right) \cdot x\right)}^{2}, 0.25, \left(-y\right) \cdot y\right) \cdot x}{\color{blue}{\left(\left(y - y \cdot y\right) \cdot 0.5\right) \cdot x - y}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 25.0%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 24.8%

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

   if -110 < x

   1. Initial program 30.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. *-rgt-identity N/A

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

      17. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -110:\\ \;\;\;\;\frac{\left(\left(-y\right) \cdot y\right) \cdot x}{\mathsf{fma}\left(0.5, x, -1\right) \cdot y} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.0% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(c, x, y)
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (c * y) * x
end function

Java

public static double code(double c, double x, double y) {
	return (c * y) * x;
}

Python

def code(c, x, y):
	return (c * y) * x

Julia

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

MATLAB

function tmp = code(c, x, y)
	tmp = (c * y) * x;
end

Wolfram

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

Derivation

1. Initial program 35.3%

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

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-lft-identity N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. log-E N/A

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

   11. log-E N/A

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

   12. metadata-eval N/A

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

   13. log-E N/A

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

   14. lower-*.f64 N/A

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

   15. log-E N/A

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

   16. *-rgt-identity N/A

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

   17. lower-*.f64 64.6

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

5. Applied rewrites 64.6%

   \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot x} \]
6. 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 2024276 
(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)))))