Logarithmic Transform

Percentage Accurate: 42.0% → 98.9%

Time: 6.2s

Alternatives: 9

Speedup: 4.9×

• could not determine a ground truth

  1. c = 6.8893567919256e+42
  2. x = 5.5860891209735885e+191
  3. y = 2.1957293483242307e+58

Specification

Math

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

FPCore

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

C

double code(double c, double x, double y) {
	return c * log((1.0 + ((pow(((double) M_E), x) - 1.0) * y)));
}

Java

public static double code(double c, double x, double y) {
	return c * Math.log((1.0 + ((Math.pow(Math.E, x) - 1.0) * y)));
}

Python

def code(c, x, y):
	return c * math.log((1.0 + ((math.pow(math.e, x) - 1.0) * y)))

Julia

function code(c, x, y)
	return Float64(c * log(Float64(1.0 + Float64(Float64((exp(1) ^ x) - 1.0) * y))))
end

MATLAB

function tmp = code(c, x, y)
	tmp = c * log((1.0 + (((2.71828182845904523536 ^ x) - 1.0) * y)));
end

Wolfram

code[c_, x_, y_] := N[(c * N[Log[N[(1.0 + N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 42.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double c, double x, double y) {
	return c * log((1.0 + ((pow(((double) M_E), x) - 1.0) * y)));
}

Java

public static double code(double c, double x, double y) {
	return c * Math.log((1.0 + ((Math.pow(Math.E, x) - 1.0) * y)));
}

Python

def code(c, x, y):
	return c * math.log((1.0 + ((math.pow(math.e, x) - 1.0) * y)))

Julia

function code(c, x, y)
	return Float64(c * log(Float64(1.0 + Float64(Float64((exp(1) ^ x) - 1.0) * y))))
end

MATLAB

function tmp = code(c, x, y)
	tmp = c * log((1.0 + (((2.71828182845904523536 ^ x) - 1.0) * y)));
end

Wolfram

code[c_, x_, y_] := N[(c * N[Log[N[(1.0 + N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-128}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* y (expm1 x))))))
   (if (<= y -1.45e-16) t_0 (if (<= y 7.5e-128) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((y * expm1(x)));
	double tmp;
	if (y <= -1.45e-16) {
		tmp = t_0;
	} else if (y <= 7.5e-128) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((y * Math.expm1(x)));
	double tmp;
	if (y <= -1.45e-16) {
		tmp = t_0;
	} else if (y <= 7.5e-128) {
		tmp = (c * Math.expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((y * math.expm1(x)))
	tmp = 0
	if y <= -1.45e-16:
		tmp = t_0
	elif y <= 7.5e-128:
		tmp = (c * math.expm1(x)) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(y * expm1(x))))
	tmp = 0.0
	if (y <= -1.45e-16)
		tmp = t_0;
	elseif (y <= 7.5e-128)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e-16], t$95$0, If[LessEqual[y, 7.5e-128], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4499999999999999e-16 or 7.50000000000000021e-128 < y

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   if -1.4499999999999999e-16 < y < 7.50000000000000021e-128

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.8

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{y} \]

      3. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left(\left(\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)\right) \cdot y, -0.5, \mathsf{expm1}\left(x\right)\right)\right) \cdot y} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-expm1.f64 77.4

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

   11. Applied rewrites 77.4%

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

Alternative 2: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ t_1 := \left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-287}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (- (pow E x) 1.0) y)) (t_1 (* (* c (expm1 x)) y)))
   (if (<= t_0 -1e-287)
     t_1
     (if (<= t_0 0.0)
       (* c (log1p (* x y)))
       (if (<= t_0 1e-12) t_1 (* (log (fma y (expm1 x) 1.0)) c))))))

C

double code(double c, double x, double y) {
	double t_0 = (pow(((double) M_E), x) - 1.0) * y;
	double t_1 = (c * expm1(x)) * y;
	double tmp;
	if (t_0 <= -1e-287) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((x * y));
	} else if (t_0 <= 1e-12) {
		tmp = t_1;
	} else {
		tmp = log(fma(y, expm1(x), 1.0)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(Float64((exp(1) ^ x) - 1.0) * y)
	t_1 = Float64(Float64(c * expm1(x)) * y)
	tmp = 0.0
	if (t_0 <= -1e-287)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(x * y)));
	elseif (t_0 <= 1e-12)
		tmp = t_1;
	else
		tmp = Float64(log(fma(y, expm1(x), 1.0)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-287], t$95$1, If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-12], t$95$1, N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -1.00000000000000002e-287 or 0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 9.9999999999999998e-13

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.8

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{y} \]

      3. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left(\left(\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)\right) \cdot y, -0.5, \mathsf{expm1}\left(x\right)\right)\right) \cdot y} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-expm1.f64 77.4

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

   11. Applied rewrites 77.4%

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

   if -1.00000000000000002e-287 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 0.0

   1. Initial program 42.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-E.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 66.1

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity 66.1

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

   6. Applied rewrites 66.1%

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

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

   1. Initial program 42.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.0

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

   3. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+222}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq -160:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -9.5e+222)
     (* (log (* (expm1 x) y)) c)
     (if (<= y -160.0) t_0 (if (<= y 1.0) (* (* c (expm1 x)) y) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -9.5e+222) {
		tmp = log((expm1(x) * y)) * c;
	} else if (y <= -160.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -9.5e+222) {
		tmp = Math.log((Math.expm1(x) * y)) * c;
	} else if (y <= -160.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (c * Math.expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -9.5e+222:
		tmp = math.log((math.expm1(x) * y)) * c
	elif y <= -160.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = (c * math.expm1(x)) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -9.5e+222)
		tmp = Float64(log(Float64(expm1(x) * y)) * c);
	elseif (y <= -160.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.5e+222], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, -160.0], t$95$0, If[LessEqual[y, 1.0], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000001e222

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around inf

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-expm1.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 5.4

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

   6. Applied rewrites 5.4%

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

   8. Applied rewrites 20.1%

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

   if -9.5000000000000001e222 < y < -160 or 1 < y

   1. Initial program 42.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-E.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 66.1

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity 66.1

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

   6. Applied rewrites 66.1%

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

   if -160 < y < 1

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.8

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{y} \]

      3. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left(\left(\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)\right) \cdot y, -0.5, \mathsf{expm1}\left(x\right)\right)\right) \cdot y} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-expm1.f64 77.4

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

   11. Applied rewrites 77.4%

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

Alternative 4: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -160:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -160.0) t_0 (if (<= y 1.0) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -160.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -160.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = (c * Math.expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -160.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = (c * math.expm1(x)) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -160.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -160.0], t$95$0, If[LessEqual[y, 1.0], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -160 or 1 < y

   1. Initial program 42.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-E.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 66.1

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity 66.1

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

   6. Applied rewrites 66.1%

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

   if -160 < y < 1

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.8

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{y} \]

      3. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left(\left(\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)\right) \cdot y, -0.5, \mathsf{expm1}\left(x\right)\right)\right) \cdot y} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-expm1.f64 77.4

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

   11. Applied rewrites 77.4%

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

Alternative 5: 80.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(x, y, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+244}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+91}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma x y 1.0)) c)))
   (if (<= y -1.02e+244) t_0 (if (<= y 1.55e+91) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(x, y, 1.0)) * c;
	double tmp;
	if (y <= -1.02e+244) {
		tmp = t_0;
	} else if (y <= 1.55e+91) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(x, y, 1.0)) * c)
	tmp = 0.0
	if (y <= -1.02e+244)
		tmp = t_0;
	elseif (y <= 1.55e+91)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(x * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1.02e+244], t$95$0, If[LessEqual[y, 1.55e+91], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e244 or 1.54999999999999999e91 < y

   1. Initial program 42.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-E.f64 40.2

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

   4. Applied rewrites 40.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 40.2

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 40.2

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-E.f64 N/A

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

      11. log-E N/A

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

      12. *-rgt-identity 40.2

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

   6. Applied rewrites 40.2%

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

   if -1.02e244 < y < 1.54999999999999999e91

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.8

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{y} \]

      3. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left(\left(\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)\right) \cdot y, -0.5, \mathsf{expm1}\left(x\right)\right)\right) \cdot y} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-expm1.f64 77.4

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

   11. Applied rewrites 77.4%

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

Alternative 6: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 7.5 \cdot 10^{+43}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 7.5e+43) (* c (* y (expm1 x))) (* (* c (expm1 x)) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 7.5e+43) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = (c * expm1(x)) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 7.5e+43) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = (c * Math.expm1(x)) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 7.5e+43:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = (c * math.expm1(x)) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 7.5e+43)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(Float64(c * expm1(x)) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 7.5e+43], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 7.49999999999999967e43

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 74.6

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

   6. Applied rewrites 74.6%

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

   if 7.49999999999999967e43 < c

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.8

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{y} \]

      3. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\left(c \cdot \mathsf{fma}\left(\left(\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)\right) \cdot y, -0.5, \mathsf{expm1}\left(x\right)\right)\right) \cdot y} \]

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 N/A

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

       2. lower-expm1.f64 77.4

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

   11. Applied rewrites 77.4%

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

Alternative 7: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-41}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(c \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -3.5e-41) (* c (* y (expm1 x))) (* x (* c y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -3.5e-41) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = x * (c * y);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -3.5e-41) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = x * (c * y);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -3.5e-41:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = x * (c * y)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -3.5e-41)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(x * Float64(c * y));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -3.5e-41], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(c * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e-41

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 74.6

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

   6. Applied rewrites 74.6%

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

   if -3.5e-41 < x

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 54.3

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, c \cdot \left(x \cdot \left(y + -1 \cdot {y}^{2}\right)\right), c \cdot y\right) \]

   6. Applied rewrites 54.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, c \cdot \left(x \cdot \left(y + -1 \cdot {y}^{2}\right)\right), c \cdot y\right)} \]

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 57.7

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-*.f64 61.6

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

   12. Applied rewrites 61.6%

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

Alternative 8: 63.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.15 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \left(c \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(c \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.15e-34) (* x (* c y)) (* y (* c x))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.15e-34) {
		tmp = x * (c * y);
	} else {
		tmp = y * (c * x);
	}
	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 <= 1.15d-34) then
        tmp = x * (c * y)
    else
        tmp = y * (c * x)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.15e-34) {
		tmp = x * (c * y);
	} else {
		tmp = y * (c * x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1.15e-34:
		tmp = x * (c * y)
	else:
		tmp = y * (c * x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.15e-34)
		tmp = Float64(x * Float64(c * y));
	else
		tmp = Float64(y * Float64(c * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1.15e-34)
		tmp = x * (c * y);
	else
		tmp = y * (c * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.15e-34], N[(x * N[(c * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(c * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.15000000000000006e-34

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-*.f64 54.3

         \[\leadsto x \cdot \mathsf{fma}\left(0.5, c \cdot \left(x \cdot \left(y + -1 \cdot {y}^{2}\right)\right), c \cdot y\right) \]

   6. Applied rewrites 54.3%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, c \cdot \left(x \cdot \left(y + -1 \cdot {y}^{2}\right)\right), c \cdot y\right)} \]

   7. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 57.7

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

   9. Applied rewrites 57.7%

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

   10. Taylor expanded in x around 0

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

       1. lower-*.f64 61.6

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

   12. Applied rewrites 61.6%

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

   if 1.15000000000000006e-34 < c

   1. Initial program 42.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 57.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 57.1

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.8

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

   3. Applied rewrites 93.8%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.8

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.8%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 59.0

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

   9. Applied rewrites 59.0%

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

Alternative 9: 61.6% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

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 = x * (c * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.0%

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lower-log1p.f64 57.1

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 57.1

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-E.f64 N/A

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

   10. e-exp-1 N/A

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

   11. pow-exp N/A

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

   12. *-lft-identity N/A

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

   13. lower-expm1.f64 93.8

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

3. Applied rewrites 93.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-*.f64 54.3

      \[\leadsto x \cdot \mathsf{fma}\left(0.5, c \cdot \left(x \cdot \left(y + -1 \cdot {y}^{2}\right)\right), c \cdot y\right) \]

6. Applied rewrites 54.3%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(0.5, c \cdot \left(x \cdot \left(y + -1 \cdot {y}^{2}\right)\right), c \cdot y\right)} \]

7. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 57.7

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

9. Applied rewrites 57.7%

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

10. Taylor expanded in x around 0

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

    1. lower-*.f64 61.6

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

12. Applied rewrites 61.6%

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

Developer Target 1: 93.8% accurate, 1.4× 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 2025156 
(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)))))