Logarithmic Transform

Percentage Accurate: 41.4% → 93.9%

Time: 4.4s

Alternatives: 7

Speedup: 5.0×

• could not determine a ground truth

  1. c = -1.5718198152181875e-249
  2. x = 4.404541241708308e+150
  3. y = 5.452515868661748e+143

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lower-log1p.f64 56.6%

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 56.6%

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-E.f64 N/A

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

   10. e-exp-1 N/A

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

   11. pow-exp N/A

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

   12. *-lft-identity N/A

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

   13. lower-expm1.f64 93.9%

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

3. Applied rewrites 93.9%

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

Alternative 2: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ t_1 := c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-288}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (let* ((t_0 (* (- (pow E x) 1.0) y)) (t_1 (* c (* y (expm1 x)))))
  (if (<= t_0 -5e-288)
    t_1
    (if (<= t_0 0.0)
      (* c (log1p (* x y)))
      (if (<= t_0 5e-31) t_1 (* (log (fma y (expm1 x) 1.0)) c))))))

C

double code(double c, double x, double y) {
	double t_0 = (pow(((double) M_E), x) - 1.0) * y;
	double t_1 = c * (y * expm1(x));
	double tmp;
	if (t_0 <= -5e-288) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((x * y));
	} else if (t_0 <= 5e-31) {
		tmp = t_1;
	} else {
		tmp = log(fma(y, expm1(x), 1.0)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(Float64((exp(1) ^ x) - 1.0) * y)
	t_1 = Float64(c * Float64(y * expm1(x)))
	tmp = 0.0
	if (t_0 <= -5e-288)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(x * y)));
	elseif (t_0 <= 5e-31)
		tmp = t_1;
	else
		tmp = Float64(log(fma(y, expm1(x), 1.0)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-288], t$95$1, If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-31], t$95$1, N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -5.0000000000000001e-288 or -0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 5.0000000000000004e-31

   1. Initial program 41.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.4%

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.5%

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

   3. Applied rewrites 51.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 74.2%

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

   6. Applied rewrites 74.2%

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

   if -5.0000000000000001e-288 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -0.0

   1. Initial program 41.4%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.6%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.6%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.9%

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

   3. Applied rewrites 93.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 66.5%

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

   6. Applied rewrites 66.5%

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

   if 5.0000000000000004e-31 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 41.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.4%

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.5%

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

   3. Applied rewrites 51.5%

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

Alternative 3: 83.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-16}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= x -4.4e-16) (* c (* y (expm1 x))) (* c (log1p (* x y)))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -4.4e-16) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -4.4e-16) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = c * Math.log1p((x * y));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -4.4e-16:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = c * math.log1p((x * y))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -4.4e-16)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -4.4e-16], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4e-16

   1. Initial program 41.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.4%

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.5%

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

   3. Applied rewrites 51.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 74.2%

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

   6. Applied rewrites 74.2%

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

   if -4.4e-16 < x

   1. Initial program 41.4%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.6%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.6%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.9%

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

   3. Applied rewrites 93.9%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 66.5%

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

   6. Applied rewrites 66.5%

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

Alternative 4: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+178}:\\ \;\;\;\;\log \left(1 + x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= y -1.1e+178)
  (* (log (+ 1.0 (* x y))) c)
  (* c (* y (expm1 x)))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.1e+178) {
		tmp = log((1.0 + (x * y))) * c;
	} else {
		tmp = c * (y * expm1(x));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.1e+178) {
		tmp = Math.log((1.0 + (x * y))) * c;
	} else {
		tmp = c * (y * Math.expm1(x));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -1.1e+178:
		tmp = math.log((1.0 + (x * y))) * c
	else:
		tmp = c * (y * math.expm1(x))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.1e+178)
		tmp = Float64(log(Float64(1.0 + Float64(x * y))) * c);
	else
		tmp = Float64(c * Float64(y * expm1(x)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.1e+178], N[(N[Log[N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e178

   1. Initial program 41.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.4%

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.5%

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

   3. Applied rewrites 51.5%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 40.3%

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

   6. Applied rewrites 40.3%

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

   if -1.1e178 < y

   1. Initial program 41.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.4%

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.5%

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

   3. Applied rewrites 51.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 74.2%

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

   6. Applied rewrites 74.2%

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

Alternative 5: 76.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-111}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= x -2e-111) (* c (* y (expm1 x))) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -2e-111) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -2e-111) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -2e-111:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -2e-111)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -2e-111], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000002e-111

   1. Initial program 41.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.4%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.4%

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.5%

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

   3. Applied rewrites 51.5%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 74.2%

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

   6. Applied rewrites 74.2%

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

   if -2.0000000000000002e-111 < x

   1. Initial program 41.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-E.f64 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-E.f64 N/A

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

      5. log-E N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.8%

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

   6. Applied rewrites 58.8%

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

Alternative 6: 64.3% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, c\right) \cdot \begin{array}{l} \mathbf{if}\;\left|c\right| \leq 8.8 \cdot 10^{+42}:\\ \;\;\;\;\left(y \cdot \left|c\right|\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left|c\right|\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (*
 (copysign 1.0 c)
 (if (<= (fabs c) 8.8e+42) (* (* y (fabs c)) x) (* (* x (fabs c)) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (fabs(c) <= 8.8e+42) {
		tmp = (y * fabs(c)) * x;
	} else {
		tmp = (x * fabs(c)) * y;
	}
	return copysign(1.0, c) * tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (Math.abs(c) <= 8.8e+42) {
		tmp = (y * Math.abs(c)) * x;
	} else {
		tmp = (x * Math.abs(c)) * y;
	}
	return Math.copySign(1.0, c) * tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if math.fabs(c) <= 8.8e+42:
		tmp = (y * math.fabs(c)) * x
	else:
		tmp = (x * math.fabs(c)) * y
	return math.copysign(1.0, c) * tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (abs(c) <= 8.8e+42)
		tmp = Float64(Float64(y * abs(c)) * x);
	else
		tmp = Float64(Float64(x * abs(c)) * y);
	end
	return Float64(copysign(1.0, c) * tmp)
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (abs(c) <= 8.8e+42)
		tmp = (y * abs(c)) * x;
	else
		tmp = (x * abs(c)) * y;
	end
	tmp_2 = (sign(c) * abs(1.0)) * tmp;
end

Wolfram

code[c_, x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[c], $MachinePrecision], 8.8e+42], N[(N[(y * N[Abs[c], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[Abs[c], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 8.8000000000000005e42

   1. Initial program 41.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-E.f64 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-E.f64 N/A

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

      5. log-E N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 61.4%

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

   6. Applied rewrites 61.4%

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

   if 8.8000000000000005e42 < c

   1. Initial program 41.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-E.f64 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-E.f64 N/A

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

      5. log-E N/A

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

      6. *-rgt-identity N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.8%

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

   6. Applied rewrites 58.8%

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

Alternative 7: 58.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower-E.f64 56.0%

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

4. Applied rewrites 56.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-E.f64 N/A

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

   5. log-E N/A

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

   6. *-rgt-identity N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 58.8%

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

6. Applied rewrites 58.8%

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

Developer Target 1: 93.9% accurate, 1.4× speedup

Math

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

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 2025234 
(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)))))