Logarithmic Transform

Percentage Accurate: 41.8% → 99.5%

Time: 30.1s

Alternatives: 8

Speedup: 19.8×

• Could not converge on a ground truth

  1. c = 6.373460277951222e-302
  2. x = 7.828587046422355e+77
  3. y = 2.1276535672431577e-141

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: 41.8% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-25} \lor \neg \left(y \leq 2.55 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -5e-25) (not (<= y 2.55e-14)))
   (* (log1p (* (expm1 x) y)) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -5e-25) || !(y <= 2.55e-14)) {
		tmp = log1p((expm1(x) * y)) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -5e-25) || !(y <= 2.55e-14)) {
		tmp = Math.log1p((Math.expm1(x) * y)) * c;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -5e-25) or not (y <= 2.55e-14):
		tmp = math.log1p((math.expm1(x) * y)) * c
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -5e-25) || !(y <= 2.55e-14))
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -5e-25], N[Not[LessEqual[y, 2.55e-14]], $MachinePrecision]], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999962e-25 or 2.5499999999999999e-14 < y

   1. Initial program 41.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.9

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

      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 44.0

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

      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 44.0

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

      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. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.6

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 99.6

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

   6. Applied rewrites 99.6%

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

   if -4.99999999999999962e-25 < y < 2.5499999999999999e-14

   1. Initial program 45.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 45.1

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

      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 69.0

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

      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 69.0

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

      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. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.2

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

   4. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.2

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 91.2

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

   6. Applied rewrites 91.2%

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

   7. Taylor expanded in y around 0

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

   9. 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 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-25} \lor \neg \left(y \leq 2.55 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -600:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{elif}\;y \leq 0.000114:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -600.0)
   (* (log1p (* y x)) c)
   (if (<= y 0.000114)
     (* (* (expm1 x) c) y)
     (* (log1p (* y (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x))) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -600.0) {
		tmp = log1p((y * x)) * c;
	} else if (y <= 0.000114) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = log1p((y * (fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -600.0)
		tmp = Float64(log1p(Float64(y * x)) * c);
	elseif (y <= 0.000114)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = Float64(log1p(Float64(y * Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x))) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -600.0], N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.000114], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(y * N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -600

   1. Initial program 59.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 59.3

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

      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 59.3

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

      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 59.3

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 57.3%

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

   if -600 < y < 1.1400000000000001e-4

   1. Initial program 45.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 45.0

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

      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 69.5

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

      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 69.5

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

      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. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.6

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

   4. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.6

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 91.6

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

   6. Applied rewrites 91.6%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 99.5%

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

   if 1.1400000000000001e-4 < y

   1. Initial program 14.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 14.4

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

      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 14.9

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

      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 14.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.7%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right)}\right) \cdot c \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -600:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{elif}\;y \leq 0.000114:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -600.0)
   (* (log1p (* y x)) c)
   (if (<= y 0.000114)
     (* (* (expm1 x) c) y)
     (* (log1p (* y (* (fma 0.5 x 1.0) x))) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -600.0) {
		tmp = log1p((y * x)) * c;
	} else if (y <= 0.000114) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = log1p((y * (fma(0.5, x, 1.0) * x))) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -600.0)
		tmp = Float64(log1p(Float64(y * x)) * c);
	elseif (y <= 0.000114)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = Float64(log1p(Float64(y * Float64(fma(0.5, x, 1.0) * x))) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -600.0], N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.000114], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(y * N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -600

   1. Initial program 59.3%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 59.3

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

      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 59.3

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

      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 59.3

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 57.3%

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

   if -600 < y < 1.1400000000000001e-4

   1. Initial program 45.0%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 45.0

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

      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 69.5

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

      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 69.5

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

      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. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.6

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

   4. Applied rewrites 91.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.6

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 91.6

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

   6. Applied rewrites 91.6%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 99.5%

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

   if 1.1400000000000001e-4 < y

   1. Initial program 14.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 14.4

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

      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 14.9

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

      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 14.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

      \[\leadsto \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)}\right) \cdot c \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -600 \lor \neg \left(y \leq 0.00021\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -600.0) (not (<= y 0.00021)))
   (* (log1p (* y x)) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -600.0) || !(y <= 0.00021)) {
		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 <= -600.0) || !(y <= 0.00021)) {
		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 <= -600.0) or not (y <= 0.00021):
		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 <= -600.0) || !(y <= 0.00021))
		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, -600.0], N[Not[LessEqual[y, 0.00021]], $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 < -600 or 2.1000000000000001e-4 < y

   1. Initial program 41.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 41.4

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

      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 41.4

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

      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 41.4

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 73.0%

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

   if -600 < y < 2.1000000000000001e-4

   1. Initial program 45.4%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 45.4

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

      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 69.6

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

      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 69.6

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.8

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

   4. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 91.8

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 91.8

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

   6. Applied rewrites 91.8%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 99.4%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -600 \lor \neg \left(y \leq 0.00021\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 10^{+74}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1e+74) (* (* (expm1 x) y) c) (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e+74) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1e+74) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1e+74:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1e+74)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1e+74], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 9.99999999999999952e73

   1. Initial program 50.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 50.1

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

      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 61.6

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

      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 61.6

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 74.4%

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

   if 9.99999999999999952e73 < c

   1. Initial program 14.2%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 14.2

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

      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 34.0

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

      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 34.0

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

      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. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 89.9

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

   4. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 89.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 89.9

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

   6. Applied rewrites 89.9%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 80.3%

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

Alternative 6: 69.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.16 \cdot 10^{-128}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.16e-128) (* (* c y) x) (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.16e-128) {
		tmp = (c * y) * x;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.16e-128) {
		tmp = (c * y) * x;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1.16e-128:
		tmp = (c * y) * x
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.16e-128)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.16e-128], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.16000000000000007e-128

   1. Initial program 50.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 69.3%

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

   if 1.16000000000000007e-128 < c

   1. Initial program 29.1%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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 29.1

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

      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 49.6

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

      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 49.6

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 93.9

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

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot c} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 93.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 93.9

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

   6. Applied rewrites 93.9%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 71.4%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 1.16 \cdot 10^{-128}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 500000:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 500000.0) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 500000.0) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 500000.0d0) 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 <= 500000.0) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 500000.0:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 500000.0)
		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 <= 500000.0)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 500000.0], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5e5

   1. Initial program 50.5%

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

   5. Applied rewrites 66.0%

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

   if 5e5 < c

   1. Initial program 18.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.2%

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

   7. Applied rewrites 55.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 500000:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.5%

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

5. Applied rewrites 62.0%

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

6. Final simplification 62.0%

   \[\leadsto \left(c \cdot y\right) \cdot x \]
7. Add Preprocessing

Developer Target 1: 94.1% 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 2025019 
(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)))))