Logarithmic Transform

Percentage Accurate: 41.5% → 99.0%

Time: 11.4s

Alternatives: 7

Speedup: 19.8×

• could not determine a ground truth

  1. c = -2.394992292375027e+140
  2. x = 9.600159074712622e+35
  3. y = 2.054234457562359e+33

Specification

Math

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

FPCore

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

Initial Program: 41.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-21} \lor \neg \left(y \leq 9.5 \cdot 10^{-78}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1.2e-21) (not (<= y 9.5e-78)))
   (* (log1p (* y (expm1 x))) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.2e-21) || !(y <= 9.5e-78)) {
		tmp = log1p((y * expm1(x))) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.2e-21) || !(y <= 9.5e-78)) {
		tmp = Math.log1p((y * Math.expm1(x))) * c;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -1.2e-21) or not (y <= 9.5e-78):
		tmp = math.log1p((y * math.expm1(x))) * c
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1.2e-21) || !(y <= 9.5e-78))
		tmp = Float64(log1p(Float64(y * expm1(x))) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1.2e-21], N[Not[LessEqual[y, 9.5e-78]], $MachinePrecision]], N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e-21 or 9.4999999999999997e-78 < y

   1. Initial program 38.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 38.1

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 45.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 45.4

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -1.2e-21 < y < 9.4999999999999997e-78

   1. Initial program 56.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.0

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 76.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.0

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. e-exp-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. e-exp-1 N/A

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

      10. exp-prod N/A

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

      11. *-lft-identity N/A

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

      12. lower-expm1.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-21} \lor \neg \left(y \leq 9.5 \cdot 10^{-78}\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65:\\ \;\;\;\;\mathsf{log1p}\left({\left(\frac{\mathsf{fma}\left(\frac{x}{y}, -0.5, {y}^{-1}\right)}{x}\right)}^{-1}\right) \cdot c\\ \mathbf{elif}\;y \leq 1.65:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.65)
   (* (log1p (pow (/ (fma (/ x y) -0.5 (pow y -1.0)) x) -1.0)) c)
   (if (<= y 1.65) (* (* (expm1 x) c) y) (* (log1p (* y x)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.65) {
		tmp = log1p(pow((fma((x / y), -0.5, pow(y, -1.0)) / x), -1.0)) * c;
	} else if (y <= 1.65) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = log1p((y * x)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.65)
		tmp = Float64(log1p((Float64(fma(Float64(x / y), -0.5, (y ^ -1.0)) / x) ^ -1.0)) * c);
	elseif (y <= 1.65)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = Float64(log1p(Float64(y * x)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.65], N[(N[Log[1 + N[Power[N[(N[(N[(x / y), $MachinePrecision] * -0.5 + N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.65], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6499999999999999

   1. Initial program 49.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 49.2

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 49.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 49.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. *-lft-identity N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-div N/A

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

      4. unpow-1 N/A

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

      5. lift-pow.f64 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 99.4

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 70.5

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

   9. Applied rewrites 70.5%

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

   if -1.6499999999999999 < y < 1.6499999999999999

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.4

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 74.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.5

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. e-exp-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. e-exp-1 N/A

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

      10. exp-prod N/A

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

      11. *-lft-identity N/A

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

      12. lower-expm1.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if 1.6499999999999999 < y

   1. Initial program 21.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 21.5

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 21.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 21.5

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. *-lft-identity N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-div N/A

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

      4. unpow-1 N/A

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

      5. lift-pow.f64 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65:\\ \;\;\;\;\mathsf{log1p}\left({\left(\frac{\mathsf{fma}\left(\frac{x}{y}, -0.5, {y}^{-1}\right)}{x}\right)}^{-1}\right) \cdot c\\ \mathbf{elif}\;y \leq 1.65:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.88:\\ \;\;\;\;\mathsf{log1p}\left(\frac{y}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}\right) \cdot c\\ \mathbf{elif}\;y \leq 1.65:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -0.88)
   (* (log1p (/ y (/ (fma -0.5 x 1.0) x))) c)
   (if (<= y 1.65) (* (* (expm1 x) c) y) (* (log1p (* y x)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -0.88) {
		tmp = log1p((y / (fma(-0.5, x, 1.0) / x))) * c;
	} else if (y <= 1.65) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = log1p((y * x)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -0.88)
		tmp = Float64(log1p(Float64(y / Float64(fma(-0.5, x, 1.0) / x))) * c);
	elseif (y <= 1.65)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = Float64(log1p(Float64(y * x)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -0.88], N[(N[Log[1 + N[(y / N[(N[(-0.5 * x + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.65], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.880000000000000004

   1. Initial program 49.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. clear-num N/A

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

      9. flip-- N/A

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

      10. lift--.f64 N/A

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

      11. inv-pow N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. lift-E.f64 N/A

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

      17. e-exp-1 N/A

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

      18. pow-exp N/A

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

      19. *-lft-identity N/A

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

      20. lower-expm1.f64 N/A

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

      21. metadata-eval 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 37.8

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

   7. Applied rewrites 37.8%

      \[\leadsto c \cdot \log \left(1 + \frac{y}{\color{blue}{\frac{\mathsf{fma}\left(-0.5, x, 1\right)}{x}}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.8

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 70.4

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

   9. Applied rewrites 70.4%

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

   if -0.880000000000000004 < y < 1.6499999999999999

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.4

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 74.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.5

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. e-exp-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. e-exp-1 N/A

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

      10. exp-prod N/A

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

      11. *-lft-identity N/A

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

      12. lower-expm1.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if 1.6499999999999999 < y

   1. Initial program 21.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 21.5

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 21.5

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 21.5

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. *-lft-identity N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-div N/A

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

      4. unpow-1 N/A

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

      5. lift-pow.f64 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.6

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

   9. Applied rewrites 99.6%

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

Alternative 4: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1700000 \lor \neg \left(y \leq 1.65\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1700000.0) (not (<= y 1.65)))
   (* (log1p (* y x)) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1700000.0) || !(y <= 1.65)) {
		tmp = log1p((y * x)) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1700000.0) || !(y <= 1.65)) {
		tmp = Math.log1p((y * x)) * c;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -1700000.0) or not (y <= 1.65):
		tmp = math.log1p((y * x)) * c
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1700000.0) || !(y <= 1.65))
		tmp = Float64(log1p(Float64(y * x)) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1700000.0], N[Not[LessEqual[y, 1.65]], $MachinePrecision]], N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e6 or 1.6499999999999999 < y

   1. Initial program 38.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 38.7

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 38.7

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 38.7

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. *-lft-identity N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-div N/A

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

      4. unpow-1 N/A

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

      5. lift-pow.f64 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow-1 N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 75.8

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

   9. Applied rewrites 75.8%

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

   if -1.7e6 < y < 1.6499999999999999

   1. Initial program 52.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 52.4

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 74.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-E.f64 N/A

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

      13. e-exp-1 N/A

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

      14. pow-exp N/A

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

      15. *-lft-identity N/A

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

      16. lower-expm1.f64 90.0

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

   4. Applied rewrites 90.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. e-exp-1 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. e-exp-1 N/A

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

      10. exp-prod N/A

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

      11. *-lft-identity N/A

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

      12. lower-expm1.f64 98.6

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

   7. Applied rewrites 98.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1700000 \lor \neg \left(y \leq 1.65\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 47.5

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

   4. lift-log.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lower-log1p.f64 61.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 61.4

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

   10. lift--.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-E.f64 N/A

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

   13. e-exp-1 N/A

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

   14. pow-exp N/A

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

   15. *-lft-identity N/A

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

   16. lower-expm1.f64 93.4

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

4. Applied rewrites 93.4%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. e-exp-1 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. e-exp-1 N/A

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

   10. exp-prod N/A

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

   11. *-lft-identity N/A

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

   12. lower-expm1.f64 79.2

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

7. Applied rewrites 79.2%

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

Alternative 6: 63.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 10^{-23}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1e-23) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e-23) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Fortran

real(8) function code(c, x, y)
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 1d-23) then
        tmp = (c * y) * x
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e-23) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1e-23:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1e-23)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1e-23)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1e-23], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 9.9999999999999996e-24

   1. Initial program 54.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. *-rgt-identity N/A

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

      17. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if 9.9999999999999996e-24 < c

   1. Initial program 26.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. *-commutative N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. metadata-eval N/A

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

      10. log-E N/A

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

      11. log-E N/A

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

      12. metadata-eval N/A

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

      13. log-E N/A

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

      14. lower-*.f64 N/A

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

      15. log-E N/A

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

      16. *-rgt-identity N/A

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

      17. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 61.0%

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

Alternative 7: 61.7% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.5%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. log-E N/A

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

   3. *-commutative N/A

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

   4. *-lft-identity N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-rgt-identity N/A

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

   9. metadata-eval N/A

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

   10. log-E N/A

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

   11. log-E N/A

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

   12. metadata-eval N/A

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

   13. log-E N/A

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

   14. lower-*.f64 N/A

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

   15. log-E N/A

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

   16. *-rgt-identity N/A

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

   17. lower-*.f64 64.4

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

5. Applied rewrites 64.4%

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

Developer Target 1: 93.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024318 
(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)))))