Logarithmic Transform

Percentage Accurate: 40.6% → 99.3%

Time: 4.7s

Alternatives: 5

Speedup: 4.9×

• could not determine a ground truth

  1. c = 6.464378708103624e-38
  2. x = 2.4756219408424196e+107
  3. y = 7.768480485917559e-284

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: 40.6% 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: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* y (expm1 x))))))
   (if (<= y -6.2e-19)
     t_0
     (if (<= y 1.6e-38)
       (* y (fma -0.5 (* c (* y (pow (expm1 x) 2.0))) (* c (expm1 x))))
       t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((y * expm1(x)));
	double tmp;
	if (y <= -6.2e-19) {
		tmp = t_0;
	} else if (y <= 1.6e-38) {
		tmp = y * fma(-0.5, (c * (y * pow(expm1(x), 2.0))), (c * expm1(x)));
	} 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 <= -6.2e-19)
		tmp = t_0;
	elseif (y <= 1.6e-38)
		tmp = Float64(y * fma(-0.5, Float64(c * Float64(y * (expm1(x) ^ 2.0))), Float64(c * expm1(x))));
	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, -6.2e-19], t$95$0, If[LessEqual[y, 1.6e-38], N[(y * N[(-0.5 * N[(c * N[(y * N[Power[N[(Exp[x] - 1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1999999999999998e-19 or 1.59999999999999989e-38 < y

   1. Initial program 40.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 55.5

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

      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 55.5

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

      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 93.6

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

   3. Applied rewrites 93.6%

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

   if -6.1999999999999998e-19 < y < 1.59999999999999989e-38

   1. Initial program 40.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 55.5

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

      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 55.5

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

      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 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-log1p.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(y \cdot \mathsf{expm1}\left(x\right) + \color{blue}{\frac{2}{2}}\right) \]

      4. add-to-fraction-rev N/A

         \[\leadsto c \cdot \log \color{blue}{\left(\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2 + 2}{2}\right)} \]

      5. lift-fma.f64 N/A

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

      6. diff-log N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 50.9

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

      10. lift-fma.f64 N/A

          \[\leadsto c \cdot \left(\log \color{blue}{\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2 + 2\right)} - \log 2\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(\log \left(\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} + 2\right) - \log 2\right) \]

      12. lift-*.f64 N/A

          \[\leadsto c \cdot \left(\log \left(2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)} + 2\right) - \log 2\right) \]

      13. associate-*r* N/A

          \[\leadsto c \cdot \left(\log \left(\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)} + 2\right) - \log 2\right) \]

      14. lower-fma.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 50.8

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

   5. Applied rewrites 50.8%

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

   7. Applied rewrites 50.7%

      \[\leadsto c \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y + y, 2\right) \cdot 4\right) - \log 8\right)} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{2}\right)\right) + c \cdot \left(e^{x} - 1\right)\right)} \]
   9. 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 76.1

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

   10. Applied rewrites 76.1%

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

Alternative 2: 93.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \end{array} \]

FPCore

(FPCore (c x y) :precision binary64 (* c (log1p (* y (expm1 x)))))

C

double code(double c, double x, double y) {
	return c * log1p((y * expm1(x)));
}

Java

public static double code(double c, double x, double y) {
	return c * Math.log1p((y * Math.expm1(x)));
}

Python

def code(c, x, y):
	return c * math.log1p((y * math.expm1(x)))

Julia

function code(c, x, y)
	return Float64(c * log1p(Float64(y * expm1(x))))
end

Wolfram

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

Derivation

1. Initial program 40.6%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
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 55.5

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

   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 55.5

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

   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 93.6

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

3. Applied rewrites 93.6%

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

Alternative 3: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (log (+ 1.0 (* (- (pow E x) 1.0) y))) 2e-14)
   (* c (* y (expm1 x)))
   (* (log (fma y (expm1 x) 1.0)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (log((1.0 + ((pow(((double) M_E), x) - 1.0) * y))) <= 2e-14) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = log(fma(y, expm1(x), 1.0)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (log(Float64(1.0 + Float64(Float64((exp(1) ^ x) - 1.0) * y))) <= 2e-14)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(log(fma(y, expm1(x), 1.0)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[N[Log[N[(1.0 + N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-14], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 #s(literal 1 binary64) (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y))) < 2e-14

   1. Initial program 40.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 55.5

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

      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 55.5

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

      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 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-log1p.f64 N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(y \cdot \mathsf{expm1}\left(x\right) + \color{blue}{\frac{2}{2}}\right) \]

      4. add-to-fraction-rev N/A

         \[\leadsto c \cdot \log \color{blue}{\left(\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2 + 2}{2}\right)} \]

      5. lift-fma.f64 N/A

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

      6. diff-log N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 50.9

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

      10. lift-fma.f64 N/A

          \[\leadsto c \cdot \left(\log \color{blue}{\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2 + 2\right)} - \log 2\right) \]

      11. *-commutative N/A

          \[\leadsto c \cdot \left(\log \left(\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} + 2\right) - \log 2\right) \]

      12. lift-*.f64 N/A

          \[\leadsto c \cdot \left(\log \left(2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)} + 2\right) - \log 2\right) \]

      13. associate-*r* N/A

          \[\leadsto c \cdot \left(\log \left(\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)} + 2\right) - \log 2\right) \]

      14. lower-fma.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 50.8

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

   5. Applied rewrites 50.8%

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

   7. Applied rewrites 50.7%

      \[\leadsto c \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y + y, 2\right) \cdot 4\right) - \log 8\right)} \]

   8. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 73.6

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

   10. Applied rewrites 73.6%

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

   if 2e-14 < (log.f64 (+.f64 #s(literal 1 binary64) (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)))

   1. Initial program 40.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 40.6

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 40.6

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.0

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

   3. Applied rewrites 51.0%

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

Alternative 4: 73.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right) \end{array} \]

FPCore

(FPCore (c x y) :precision binary64 (* c (* y (expm1 x))))

C

double code(double c, double x, double y) {
	return c * (y * expm1(x));
}

Java

public static double code(double c, double x, double y) {
	return c * (y * Math.expm1(x));
}

Python

def code(c, x, y):
	return c * (y * math.expm1(x))

Julia

function code(c, x, y)
	return Float64(c * Float64(y * expm1(x)))
end

Wolfram

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

Derivation

1. Initial program 40.6%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
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 55.5

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

   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 55.5

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

   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 93.6

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

3. Applied rewrites 93.6%

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

   1. lift-log1p.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

      \[\leadsto c \cdot \log \left(y \cdot \mathsf{expm1}\left(x\right) + \color{blue}{\frac{2}{2}}\right) \]

   4. add-to-fraction-rev N/A

      \[\leadsto c \cdot \log \color{blue}{\left(\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2 + 2}{2}\right)} \]

   5. lift-fma.f64 N/A

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

   6. diff-log N/A

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

   7. lift-log.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 50.9

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

   10. lift-fma.f64 N/A

       \[\leadsto c \cdot \left(\log \color{blue}{\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2 + 2\right)} - \log 2\right) \]

   11. *-commutative N/A

       \[\leadsto c \cdot \left(\log \left(\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} + 2\right) - \log 2\right) \]

   12. lift-*.f64 N/A

       \[\leadsto c \cdot \left(\log \left(2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)} + 2\right) - \log 2\right) \]

   13. associate-*r* N/A

       \[\leadsto c \cdot \left(\log \left(\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)} + 2\right) - \log 2\right) \]

   14. lower-fma.f64 N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f64 50.8

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

5. Applied rewrites 50.8%

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

7. Applied rewrites 50.7%

   \[\leadsto c \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y + y, 2\right) \cdot 4\right) - \log 8\right)} \]

8. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-expm1.f64 73.6

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

10. Applied rewrites 73.6%

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

Alternative 5: 55.7% accurate, 4.9× speedup

Math

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

FPCore

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

C

double code(double c, double x, double y) {
	return c * (x * 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 = c * (x * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.6%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
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 55.5

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

   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 55.5

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

   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 93.6

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

3. Applied rewrites 93.6%

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

   1. lift-log1p.f64 N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

      \[\leadsto c \cdot \log \left(y \cdot \mathsf{expm1}\left(x\right) + \color{blue}{\frac{2}{2}}\right) \]

   4. add-to-fraction-rev N/A

      \[\leadsto c \cdot \log \color{blue}{\left(\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2 + 2}{2}\right)} \]

   5. lift-fma.f64 N/A

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

   6. diff-log N/A

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

   7. lift-log.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 50.9

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

   10. lift-fma.f64 N/A

       \[\leadsto c \cdot \left(\log \color{blue}{\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2 + 2\right)} - \log 2\right) \]

   11. *-commutative N/A

       \[\leadsto c \cdot \left(\log \left(\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} + 2\right) - \log 2\right) \]

   12. lift-*.f64 N/A

       \[\leadsto c \cdot \left(\log \left(2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)} + 2\right) - \log 2\right) \]

   13. associate-*r* N/A

       \[\leadsto c \cdot \left(\log \left(\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)} + 2\right) - \log 2\right) \]

   14. lower-fma.f64 N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f64 50.8

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

5. Applied rewrites 50.8%

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

7. Applied rewrites 50.7%

   \[\leadsto c \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y + y, 2\right) \cdot 4\right) - \log 8\right)} \]

8. Taylor expanded in x around 0

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

   1. lower-*.f64 55.7

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

10. Applied rewrites 55.7%

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

Developer Target 1: 93.6% 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 2025164 
(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)))))