Logarithmic Transform

Percentage Accurate: 42.1% → 99.3%

Time: 33.8s

Alternatives: 8

Speedup: 5.0×

• could not determine a ground truth

  1. c = 4.44155967131951e+165
  2. x = 1.3135589745223551e+93
  3. y = 1.0583433478944147e+276

Specification

Math

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

FPCore

(FPCore (c x y)
  :precision binary64
  :pre TRUE
  (* 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]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	c * (ln(((1) + (((exp(1) ^ x) - (1)) * y))))
END code

Initial Program: 42.1% 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
  :pre TRUE
  (* 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]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	c * (ln(((1) + (((exp(1) ^ x) - (1)) * y))))
END code

Alternative 1: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -1.5511526834195688 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.106409269880449 \cdot 10^{-55}:\\ \;\;\;\;y \cdot \left(c \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
  (if (<= y -1.5511526834195688e-28)
    t_0
    (if (<= y 1.106409269880449e-55) (* y (* c (expm1 x))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -1.5511526834195688e-28) {
		tmp = t_0;
	} else if (y <= 1.106409269880449e-55) {
		tmp = y * (c * expm1(x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((Math.expm1(x) * y));
	double tmp;
	if (y <= -1.5511526834195688e-28) {
		tmp = t_0;
	} else if (y <= 1.106409269880449e-55) {
		tmp = y * (c * Math.expm1(x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((math.expm1(x) * y))
	tmp = 0
	if y <= -1.5511526834195688e-28:
		tmp = t_0
	elif y <= 1.106409269880449e-55:
		tmp = y * (c * math.expm1(x))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(expm1(x) * y)))
	tmp = 0.0
	if (y <= -1.5511526834195688e-28)
		tmp = t_0;
	elseif (y <= 1.106409269880449e-55)
		tmp = Float64(y * Float64(c * expm1(x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5511526834195688e-28], t$95$0, If[LessEqual[y, 1.106409269880449e-55], N[(y * N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	LET t_0 = (c * (ln(((((exp(x)) - (1)) * y) + (1))))) IN
		LET tmp_1 = IF (y <= (1106409269880448986171765822828396570300838916253403248766447726993267927555191844216655416972111226111754746531116047854664735489932931417587924638468166449456475675106048583984375e-235)) THEN (y * (c * ((exp(x)) - (1)))) ELSE t_0 ENDIF IN
		LET tmp = IF (y <= (-1551152683419568771045439616240171027412485134580889752304752284356722060766620163718698677257634699344635009765625e-142)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < -1.5511526834195688e-28 or 1.106409269880449e-55 < y

   1. Initial program 42.1%

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

   3. Applied rewrites 93.3%

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

   if -1.5511526834195688e-28 < y < 1.106409269880449e-55

   1. Initial program 42.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 74.1%

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

   6. Applied rewrites 77.6%

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

Alternative 2: 88.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right)\\ \mathbf{if}\;y \leq -5.756465638514478 \cdot 10^{+95}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.6993600723541415 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(c \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* c (log (fma (expm1 x) y 1.0)))))
  (if (<= y -5.756465638514478e+95)
    t_0
    (if (<= y 6.6993600723541415e+153) (* y (* c (expm1 x))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log(fma(expm1(x), y, 1.0));
	double tmp;
	if (y <= -5.756465638514478e+95) {
		tmp = t_0;
	} else if (y <= 6.6993600723541415e+153) {
		tmp = y * (c * expm1(x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(fma(expm1(x), y, 1.0)))
	tmp = 0.0
	if (y <= -5.756465638514478e+95)
		tmp = t_0;
	elseif (y <= 6.6993600723541415e+153)
		tmp = Float64(y * Float64(c * expm1(x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.756465638514478e+95], t$95$0, If[LessEqual[y, 6.6993600723541415e+153], N[(y * N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	LET t_0 = (c * (ln(((((exp(x)) - (1)) * y) + (1))))) IN
		LET tmp_1 = IF (y <= (6699360072354141475203966126958235020504119986664746420655450416218605553402243332468594199816495235077034820225283496654424754774302170343137135535587328)) THEN (y * (c * ((exp(x)) - (1)))) ELSE t_0 ENDIF IN
		LET tmp = IF (y <= (-575646563851447818334308706974014366108583700322314004965891818191341923109222785033876646395904)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < -5.7564656385144782e95 or 6.6993600723541415e153 < y

   1. Initial program 42.1%

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

   3. Applied rewrites 51.5%

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

   if -5.7564656385144782e95 < y < 6.6993600723541415e153

   1. Initial program 42.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 74.1%

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

   6. Applied rewrites 77.6%

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

Alternative 3: 79.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 6.6993600723541415 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \left(c \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \log \left(1 + x \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  :pre TRUE
  (if (<= y 6.6993600723541415e+153)
  (* y (* c (expm1 x)))
  (* c (log (+ 1.0 (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 6.6993600723541415e+153) {
		tmp = y * (c * expm1(x));
	} else {
		tmp = c * log((1.0 + (x * y)));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= 6.6993600723541415e+153) {
		tmp = y * (c * Math.expm1(x));
	} else {
		tmp = c * Math.log((1.0 + (x * y)));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= 6.6993600723541415e+153:
		tmp = y * (c * math.expm1(x))
	else:
		tmp = c * math.log((1.0 + (x * y)))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 6.6993600723541415e+153)
		tmp = Float64(y * Float64(c * expm1(x)));
	else
		tmp = Float64(c * log(Float64(1.0 + Float64(x * y))));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 6.6993600723541415e+153], N[(y * N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	LET tmp = IF (y <= (6699360072354141475203966126958235020504119986664746420655450416218605553402243332468594199816495235077034820225283496654424754774302170343137135535587328)) THEN (y * (c * ((exp(x)) - (1)))) ELSE (c * (ln(((1) + (x * y))))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < 6.6993600723541415e153

   1. Initial program 42.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 74.1%

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

   6. Applied rewrites 77.6%

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

   if 6.6993600723541415e153 < y

   1. Initial program 42.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 40.1%

      \[\leadsto c \cdot \log \left(1 + x \cdot y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -7.417339803987706 \cdot 10^{-58}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(c \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  :pre TRUE
  (if (<= x -7.417339803987706e-58) (* c (* y (expm1 x))) (* x (* c y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -7.417339803987706e-58) {
		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 <= -7.417339803987706e-58) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = x * (c * y);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -7.417339803987706e-58:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = x * (c * y)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -7.417339803987706e-58)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(x * Float64(c * y));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -7.417339803987706e-58], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(c * y), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	LET tmp = IF (x <= (-7417339803987706051609638428185765246820801495451024508716982989045805325943245960443833233383756116685077234171247789697125728316515361223759499054164479048267821781337261199951171875e-241)) THEN (c * (y * ((exp(x)) - (1)))) ELSE (x * (c * y)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < -7.4173398039877061e-58

   1. Initial program 42.1%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 74.1%

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

   if -7.4173398039877061e-58 < x

   1. Initial program 42.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 56.2%

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

   6. Applied rewrites 62.3%

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

Alternative 5: 77.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	y * (c * ((exp(x)) - (1)))
END code

Derivation

1. Initial program 42.1%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 74.1%

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

6. Applied rewrites 77.6%

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

Alternative 6: 62.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	x * (c * y)
END code

Derivation

1. Initial program 42.1%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 56.2%

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

6. Applied rewrites 62.3%

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

Alternative 7: 58.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -5.742794795096544 \cdot 10^{+167}:\\ \;\;\;\;0 \cdot c\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  :pre TRUE
  (if (<= x -5.742794795096544e+167) (* 0.0 c) (* c (* x y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -5.742794795096544e+167) {
		tmp = 0.0 * c;
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Fortran

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 (x <= (-5.742794795096544d+167)) then
        tmp = 0.0d0 * c
    else
        tmp = c * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -5.742794795096544e+167) {
		tmp = 0.0 * c;
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -5.742794795096544e+167:
		tmp = 0.0 * c
	else:
		tmp = c * (x * y)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -5.742794795096544e+167)
		tmp = Float64(0.0 * c);
	else
		tmp = Float64(c * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (x <= -5.742794795096544e+167)
		tmp = 0.0 * c;
	else
		tmp = c * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -5.742794795096544e+167], N[(0.0 * c), $MachinePrecision], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	LET tmp = IF (x <= (-574279479509654428513738203990391290412983179384704553627057515023638265121933065921031923646839306755009345843997374291291218962910957683421804481399581996264607186944)) THEN ((0) * c) ELSE (c * (x * y)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < -5.7427947950965443e167

   1. Initial program 42.1%

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

   3. Applied rewrites 51.4%

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

   5. Applied rewrites 25.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \frac{1}{2} + c \cdot \log 2 \]

   8. Applied rewrites 5.9%

      \[\leadsto \mathsf{fma}\left(c, \log 0.5, c \cdot \log 2\right) \]

   10. Applied rewrites 31.2%

       \[\leadsto 0 \cdot c \]

   if -5.7427947950965443e167 < x

   1. Initial program 42.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 56.2%

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

Alternative 8: 31.2% accurate, 8.8× speedup

Math

\[0 \cdot c \]

FPCore

(FPCore (c x y)
  :precision binary64
  :pre TRUE
  (* 0.0 c))

C

double code(double c, double x, double y) {
	return 0.0 * c;
}

Fortran

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 = 0.0d0 * c
end function

Java

public static double code(double c, double x, double y) {
	return 0.0 * c;
}

Python

def code(c, x, y):
	return 0.0 * c

Julia

function code(c, x, y)
	return Float64(0.0 * c)
end

MATLAB

function tmp = code(c, x, y)
	tmp = 0.0 * c;
end

Wolfram

code[c_, x_, y_] := N[(0.0 * c), $MachinePrecision]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	(0) * c
END code

Derivation

1. Initial program 42.1%

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

3. Applied rewrites 51.4%

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

5. Applied rewrites 25.6%

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

6. Taylor expanded in x around 0

   \[\leadsto c \cdot \log \frac{1}{2} + c \cdot \log 2 \]

8. Applied rewrites 5.9%

   \[\leadsto \mathsf{fma}\left(c, \log 0.5, c \cdot \log 2\right) \]

10. Applied rewrites 31.2%

    \[\leadsto 0 \cdot c \]
11. Add Preprocessing

Developer Target 1: 93.3% 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
  :pre TRUE
  (* 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]

ReFlow

f(c, x, y):
	c in [-inf, +inf],
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(c, x, y: real): real =
	c * (ln(((((exp(x)) - (1)) * y) + (1))))
END code

Reproduce

herbie shell --seed 2026089 +o generate:egglog
(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)))))