Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B

Percentage Accurate: 99.8% → 99.8%

Time: 11.6s

Alternatives: 20

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 22.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;i \cdot y\\ \mathbf{elif}\;t\_1 \leq -100:\\ \;\;\;\;\frac{z}{i} \cdot i\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \mathbf{else}:\\ \;\;\;\;i \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (<= t_1 -2e+306)
     (* i y)
     (if (<= t_1 -100.0)
       (* (/ z i) i)
       (if (<= t_1 5e+307) (* (/ a i) i) (* i y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = i * y;
	} else if (t_1 <= -100.0) {
		tmp = (z / i) * i;
	} else if (t_1 <= 5e+307) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if (t_1 <= (-2d+306)) then
        tmp = i * y
    else if (t_1 <= (-100.0d0)) then
        tmp = (z / i) * i
    else if (t_1 <= 5d+307) then
        tmp = (a / i) * i
    else
        tmp = i * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = i * y;
	} else if (t_1 <= -100.0) {
		tmp = (z / i) * i;
	} else if (t_1 <= 5e+307) {
		tmp = (a / i) * i;
	} else {
		tmp = i * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if t_1 <= -2e+306:
		tmp = i * y
	elif t_1 <= -100.0:
		tmp = (z / i) * i
	elif t_1 <= 5e+307:
		tmp = (a / i) * i
	else:
		tmp = i * y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(i * y);
	elseif (t_1 <= -100.0)
		tmp = Float64(Float64(z / i) * i);
	elseif (t_1 <= 5e+307)
		tmp = Float64(Float64(a / i) * i);
	else
		tmp = Float64(i * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if (t_1 <= -2e+306)
		tmp = i * y;
	elseif (t_1 <= -100.0)
		tmp = (z / i) * i;
	elseif (t_1 <= 5e+307)
		tmp = (a / i) * i;
	else
		tmp = i * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+306], N[(i * y), $MachinePrecision], If[LessEqual[t$95$1, -100.0], N[(N[(z / i), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[t$95$1, 5e+307], N[(N[(a / i), $MachinePrecision] * i), $MachinePrecision], N[(i * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -2.00000000000000003e306 or 5e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 97.3%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 86.9

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 86.9%

      \[\leadsto \color{blue}{i \cdot y} \]

   if -2.00000000000000003e306 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -100

   1. Initial program 99.8%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(y + \left(\frac{a}{i} + \left(\frac{t}{i} + \left(\frac{z}{i} + \left(\frac{x \cdot \log y}{i} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{i}\right)\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. div-add-rev N/A

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

      3. div-add-rev N/A

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

      4. div-add N/A

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

      5. div-add N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{z}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 9.5%

      \[\leadsto \frac{z}{i} \cdot i \]

   if -100 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5e307

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(y + \left(\frac{a}{i} + \left(\frac{t}{i} + \left(\frac{z}{i} + \left(\frac{x \cdot \log y}{i} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{i}\right)\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. div-add-rev N/A

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

      3. div-add-rev N/A

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

      4. div-add N/A

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

      5. div-add N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 11.9%

      \[\leadsto \frac{a}{i} \cdot i \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 22.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+26} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (+
          (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c)))
          (* y i))))
   (if (or (<= t_1 5e+26) (not (<= t_1 5e+307))) (* i y) (* (/ a i) i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	double tmp;
	if ((t_1 <= 5e+26) || !(t_1 <= 5e+307)) {
		tmp = i * y;
	} else {
		tmp = (a / i) * i;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5d0) * log(c))) + (y * i)
    if ((t_1 <= 5d+26) .or. (.not. (t_1 <= 5d+307))) then
        tmp = i * y
    else
        tmp = (a / i) * i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (((((x * Math.log(y)) + z) + t) + a) + ((b - 0.5) * Math.log(c))) + (y * i);
	double tmp;
	if ((t_1 <= 5e+26) || !(t_1 <= 5e+307)) {
		tmp = i * y;
	} else {
		tmp = (a / i) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (((((x * math.log(y)) + z) + t) + a) + ((b - 0.5) * math.log(c))) + (y * i)
	tmp = 0
	if (t_1 <= 5e+26) or not (t_1 <= 5e+307):
		tmp = i * y
	else:
		tmp = (a / i) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i))
	tmp = 0.0
	if ((t_1 <= 5e+26) || !(t_1 <= 5e+307))
		tmp = Float64(i * y);
	else
		tmp = Float64(Float64(a / i) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i);
	tmp = 0.0;
	if ((t_1 <= 5e+26) || ~((t_1 <= 5e+307)))
		tmp = i * y;
	else
		tmp = (a / i) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 5e+26], N[Not[LessEqual[t$95$1, 5e+307]], $MachinePrecision]], N[(i * y), $MachinePrecision], N[(N[(a / i), $MachinePrecision] * i), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5.0000000000000001e26 or 5e307 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.2%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{i \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 30.3

         \[\leadsto \color{blue}{i \cdot y} \]

   5. Applied rewrites 30.3%

      \[\leadsto \color{blue}{i \cdot y} \]

   if 5.0000000000000001e26 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < 5e307

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(y + \left(\frac{a}{i} + \left(\frac{t}{i} + \left(\frac{z}{i} + \left(\frac{x \cdot \log y}{i} + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{i}\right)\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. div-add-rev N/A

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

      3. div-add-rev N/A

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

      4. div-add N/A

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

      5. div-add N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 64.8%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a}{i} \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 12.3%

      \[\leadsto \frac{a}{i} \cdot i \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 5 \cdot 10^{+26} \lor \neg \left(\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;i \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{i} \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\\ \mathbf{if}\;\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \leq -5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, t\_1\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (log y) x (fma (- b 0.5) (log c) t))))
   (if (<=
        (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i))
        -5e+22)
     (+ (fma i y z) t_1)
     (+ (fma i y t_1) a))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(log(y), x, fma((b - 0.5), log(c), t));
	double tmp;
	if (((((((x * log(y)) + z) + t) + a) + ((b - 0.5) * log(c))) + (y * i)) <= -5e+22) {
		tmp = fma(i, y, z) + t_1;
	} else {
		tmp = fma(i, y, t_1) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(log(y), x, fma(Float64(b - 0.5), log(c), t))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(Float64(x * log(y)) + z) + t) + a) + Float64(Float64(b - 0.5) * log(c))) + Float64(y * i)) <= -5e+22)
		tmp = Float64(fma(i, y, z) + t_1);
	else
		tmp = Float64(fma(i, y, t_1) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], -5e+22], N[(N[(i * y + z), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(i * y + t$95$1), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i)) < -4.9999999999999996e22

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 82.8

          \[\leadsto \mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, t\right)\right) \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)} \]

   if -4.9999999999999996e22 < (+.f64 (+.f64 (+.f64 (+.f64 (+.f64 (*.f64 x (log.f64 y)) z) t) a) (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))) (*.f64 y i))

   1. Initial program 99.2%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\right) + a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.05 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-0.5, \log c, t\right)\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= a 2.05e+135)
   (+ (fma i y z) (fma (log y) x (fma (- b 0.5) (log c) t)))
   (+ (fma i y (fma (log y) x (fma -0.5 (log c) t))) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (a <= 2.05e+135) {
		tmp = fma(i, y, z) + fma(log(y), x, fma((b - 0.5), log(c), t));
	} else {
		tmp = fma(i, y, fma(log(y), x, fma(-0.5, log(c), t))) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (a <= 2.05e+135)
		tmp = Float64(fma(i, y, z) + fma(log(y), x, fma(Float64(b - 0.5), log(c), t)));
	else
		tmp = Float64(fma(i, y, fma(log(y), x, fma(-0.5, log(c), t))) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[a, 2.05e+135], N[(N[(i * y + z), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * y + N[(N[Log[y], $MachinePrecision] * x + N[(-0.5 * N[Log[c], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.05e135

   1. Initial program 99.4%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower--.f64 N/A

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

      16. lower-log.f64 89.7

          \[\leadsto \mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, t\right)\right) \]

   5. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, z\right) + \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)} \]

   if 2.05e135 < a

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\right) + a} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, t + \frac{-1}{2} \cdot \log c\right)\right) + a \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-0.5, \log c, t\right)\right)\right) + a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{+29}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, t\right)\right) + z\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 4.7e+29)
   (+ (+ (fma (log y) x (fma (log c) (- b 0.5) t)) z) a)
   (+ (+ a t) (fma (- b 0.5) (log c) (fma i y z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 4.7e+29) {
		tmp = (fma(log(y), x, fma(log(c), (b - 0.5), t)) + z) + a;
	} else {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(i, y, z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 4.7e+29)
		tmp = Float64(Float64(fma(log(y), x, fma(log(c), Float64(b - 0.5), t)) + z) + a);
	else
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(i, y, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 4.7e+29], N[(N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.7000000000000002e29

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 80.5

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 80.5%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.5%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 96.8

          \[\leadsto \left(\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, \color{blue}{b - 0.5}, t\right)\right) + z\right) + a \]

   10. Applied rewrites 96.8%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\log c, b - 0.5, t\right)\right) + z\right) + a} \]

   if 4.7000000000000002e29 < y

   1. Initial program 98.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 88.8

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(\log c, b - 0.5, \log y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 9e+208)
   (fma (- b 0.5) (log c) (+ (fma i y z) (+ a t)))
   (+ (+ a t) (fma (log c) (- b 0.5) (* (log y) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 9e+208) {
		tmp = fma((b - 0.5), log(c), (fma(i, y, z) + (a + t)));
	} else {
		tmp = (a + t) + fma(log(c), (b - 0.5), (log(y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 9e+208)
		tmp = fma(Float64(b - 0.5), log(c), Float64(fma(i, y, z) + Float64(a + t)));
	else
		tmp = Float64(Float64(a + t) + fma(log(c), Float64(b - 0.5), Float64(log(y) * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 9e+208], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(i * y + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + t), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.00000000000000029e208

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.0

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   if 9.00000000000000029e208 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\right) + a} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.3%

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

Alternative 8: 87.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(\log y, x, t\right)\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 9e+208)
   (fma (- b 0.5) (log c) (+ (fma i y z) (+ a t)))
   (+ (fma -0.5 (log c) (fma (log y) x t)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 9e+208) {
		tmp = fma((b - 0.5), log(c), (fma(i, y, z) + (a + t)));
	} else {
		tmp = fma(-0.5, log(c), fma(log(y), x, t)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 9e+208)
		tmp = fma(Float64(b - 0.5), log(c), Float64(fma(i, y, z) + Float64(a + t)));
	else
		tmp = Float64(fma(-0.5, log(c), fma(log(y), x, t)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 9e+208], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(i * y + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[c], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * x + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.00000000000000029e208

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.0

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   if 9.00000000000000029e208 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(b - 0.5, \log c, t\right)\right)\right) + a} \]

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 80.3%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 76.6%

       \[\leadsto \mathsf{fma}\left(-0.5, \log c, \mathsf{fma}\left(\log y, x, t\right)\right) + a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(b - 0.5\right) \cdot \log c \leq -1 \cdot 10^{+273}:\\ \;\;\;\;\log c \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot z + t\right) + a\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= (* (- b 0.5) (log c)) -1e+273)
   (* (log c) b)
   (+ (+ (+ (* 1.0 z) t) a) (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((b - 0.5) * log(c)) <= -1e+273) {
		tmp = log(c) * b;
	} else {
		tmp = (((1.0 * z) + t) + a) + (y * i);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (((b - 0.5d0) * log(c)) <= (-1d+273)) then
        tmp = log(c) * b
    else
        tmp = (((1.0d0 * z) + t) + a) + (y * i)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((b - 0.5) * Math.log(c)) <= -1e+273) {
		tmp = Math.log(c) * b;
	} else {
		tmp = (((1.0 * z) + t) + a) + (y * i);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((b - 0.5) * math.log(c)) <= -1e+273:
		tmp = math.log(c) * b
	else:
		tmp = (((1.0 * z) + t) + a) + (y * i)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(b - 0.5) * log(c)) <= -1e+273)
		tmp = Float64(log(c) * b);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 * z) + t) + a) + Float64(y * i));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (((b - 0.5) * log(c)) <= -1e+273)
		tmp = log(c) * b;
	else
		tmp = (((1.0 * z) + t) + a) + (y * i);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision]), $MachinePrecision], -1e+273], N[(N[Log[c], $MachinePrecision] * b), $MachinePrecision], N[(N[(N[(N[(1.0 * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c)) < -9.99999999999999945e272

   1. Initial program 99.3%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \log c} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log c \cdot b} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log c \cdot b} \]

      3. lower-log.f64 81.6

         \[\leadsto \color{blue}{\log c} \cdot b \]

   5. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\log c \cdot b} \]

   if -9.99999999999999945e272 < (*.f64 (-.f64 b #s(literal 1/2 binary64)) (log.f64 c))

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(a + \left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} + y \cdot i \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + t\right) + a\right) + y \cdot i \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + z\right) + t\right) + a\right) + y \cdot i \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 83.6

         \[\leadsto \left(\left(\mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, z\right) + t\right) + a\right) + y \cdot i \]

   5. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right)} + y \cdot i \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right) + t\right) + a\right) + y \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 73.8%

      \[\leadsto \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z + t\right) + a\right) + y \cdot i \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
   10. Step-by-step derivation

   11. Applied rewrites 69.3%

       \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 1.5e+212)
   (fma (- b 0.5) (log c) (+ (fma i y z) (+ a t)))
   (* (log y) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 1.5e+212) {
		tmp = fma((b - 0.5), log(c), (fma(i, y, z) + (a + t)));
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 1.5e+212)
		tmp = fma(Float64(b - 0.5), log(c), Float64(fma(i, y, z) + Float64(a + t)));
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 1.5e+212], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(i * y + z), $MachinePrecision] + N[(a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e212

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.0

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   if 1.5e212 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 38.5

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      3. lower-log.f64 63.9

         \[\leadsto \color{blue}{\log y} \cdot x \]

   8. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+212}:\\ \;\;\;\;\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 1.5e+212)
   (+ (+ a t) (fma (- b 0.5) (log c) (fma i y z)))
   (* (log y) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 1.5e+212) {
		tmp = (a + t) + fma((b - 0.5), log(c), fma(i, y, z));
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 1.5e+212)
		tmp = Float64(Float64(a + t) + fma(Float64(b - 0.5), log(c), fma(i, y, z)));
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 1.5e+212], N[(N[(a + t), $MachinePrecision] + N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(i * y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e212

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.0

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]

   if 1.5e212 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 38.5

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      3. lower-log.f64 63.9

         \[\leadsto \color{blue}{\log y} \cdot x \]

   8. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 71.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+212}:\\ \;\;\;\;\left(a + z\right) + \mathsf{fma}\left(\log c, b - 0.5, i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 1.5e+212)
   (+ (+ a z) (fma (log c) (- b 0.5) (* i y)))
   (* (log y) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 1.5e+212) {
		tmp = (a + z) + fma(log(c), (b - 0.5), (i * y));
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 1.5e+212)
		tmp = Float64(Float64(a + z) + fma(log(c), Float64(b - 0.5), Float64(i * y)));
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 1.5e+212], N[(N[(a + z), $MachinePrecision] + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e212

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.0

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto a + \color{blue}{\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.0%

      \[\leadsto \left(a + z\right) + \color{blue}{\mathsf{fma}\left(\log c, b - 0.5, i \cdot y\right)} \]

   if 1.5e212 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 38.5

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      3. lower-log.f64 63.9

         \[\leadsto \color{blue}{\log y} \cdot x \]

   8. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+212}:\\ \;\;\;\;\left(a + z\right) + \mathsf{fma}\left(\log c, b - 0.5, i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+212}:\\ \;\;\;\;\mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + a\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 1.5e+212) (+ (fma i y (fma (log c) (- b 0.5) z)) a) (* (log y) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 1.5e+212) {
		tmp = fma(i, y, fma(log(c), (b - 0.5), z)) + a;
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 1.5e+212)
		tmp = Float64(fma(i, y, fma(log(c), Float64(b - 0.5), z)) + a);
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 1.5e+212], N[(N[(i * y + N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e212

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 89.0

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.0%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   8. Taylor expanded in t around 0

      \[\leadsto a + \color{blue}{\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.0%

       \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + \color{blue}{a} \]

   if 1.5e212 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 38.5

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      3. lower-log.f64 63.9

         \[\leadsto \color{blue}{\log y} \cdot x \]

   8. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 61.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+85}:\\ \;\;\;\;\left(\left(1 \cdot z + t\right) + a\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, i \cdot y\right) + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= z -1.35e+85)
   (+ (+ (+ (* 1.0 z) t) a) (* y i))
   (+ (fma (log c) (- b 0.5) (* i y)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (z <= -1.35e+85) {
		tmp = (((1.0 * z) + t) + a) + (y * i);
	} else {
		tmp = fma(log(c), (b - 0.5), (i * y)) + a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (z <= -1.35e+85)
		tmp = Float64(Float64(Float64(Float64(1.0 * z) + t) + a) + Float64(y * i));
	else
		tmp = Float64(fma(log(c), Float64(b - 0.5), Float64(i * y)) + a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[z, -1.35e+85], N[(N[(N[(N[(1.0 * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.34999999999999992e85

   1. Initial program 100.0%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(a + \left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} + y \cdot i \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + t\right) + a\right) + y \cdot i \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + z\right) + t\right) + a\right) + y \cdot i \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 94.9

         \[\leadsto \left(\left(\mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, z\right) + t\right) + a\right) + y \cdot i \]

   5. Applied rewrites 94.9%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right)} + y \cdot i \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right) + t\right) + a\right) + y \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 94.9%

      \[\leadsto \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z + t\right) + a\right) + y \cdot i \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
   10. Step-by-step derivation

   11. Applied rewrites 75.5%

       \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]

   if -1.34999999999999992e85 < z

   1. Initial program 99.4%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 82.2

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 82.2%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 82.2%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   8. Taylor expanded in t around 0

      \[\leadsto a + \color{blue}{\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + \color{blue}{a} \]

   11. Taylor expanded in z around 0

       \[\leadsto \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right) + a \]
   12. Step-by-step derivation

   13. Applied rewrites 54.9%

       \[\leadsto \mathsf{fma}\left(\log c, b - 0.5, i \cdot y\right) + a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(b - 0.5, \log c, \left(a + t\right) + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot z + t\right) + a\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 5.5e+31)
   (fma (- b 0.5) (log c) (+ (+ a t) z))
   (+ (+ (+ (* 1.0 z) t) a) (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 5.5e+31) {
		tmp = fma((b - 0.5), log(c), ((a + t) + z));
	} else {
		tmp = (((1.0 * z) + t) + a) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 5.5e+31)
		tmp = fma(Float64(b - 0.5), log(c), Float64(Float64(a + t) + z));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 * z) + t) + a) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 5.5e+31], N[(N[(b - 0.5), $MachinePrecision] * N[Log[c], $MachinePrecision] + N[(N[(a + t), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.50000000000000002e31

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 80.6

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.6%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto a + \color{blue}{\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.4%

       \[\leadsto \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + t\right) + \color{blue}{a} \]
   11. Step-by-step derivation

   12. Applied rewrites 77.4%

       \[\leadsto \mathsf{fma}\left(b - 0.5, \log c, \left(a + t\right) + z\right) \]

   if 5.50000000000000002e31 < y

   1. Initial program 98.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(a + \left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} + y \cdot i \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + t\right) + a\right) + y \cdot i \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + z\right) + t\right) + a\right) + y \cdot i \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 88.7

         \[\leadsto \left(\left(\mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, z\right) + t\right) + a\right) + y \cdot i \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right)} + y \cdot i \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right) + t\right) + a\right) + y \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 78.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z + t\right) + a\right) + y \cdot i \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
   10. Step-by-step derivation

   11. Applied rewrites 78.1%

       \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + t\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot z + t\right) + a\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 5.5e+31)
   (+ (+ (fma (log c) (- b 0.5) z) t) a)
   (+ (+ (+ (* 1.0 z) t) a) (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 5.5e+31) {
		tmp = (fma(log(c), (b - 0.5), z) + t) + a;
	} else {
		tmp = (((1.0 * z) + t) + a) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 5.5e+31)
		tmp = Float64(Float64(fma(log(c), Float64(b - 0.5), z) + t) + a);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 * z) + t) + a) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 5.5e+31], N[(N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision], N[(N[(N[(N[(1.0 * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.50000000000000002e31

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 80.6

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.6%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   8. Taylor expanded in y around 0

      \[\leadsto a + \color{blue}{\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.4%

       \[\leadsto \left(\mathsf{fma}\left(\log c, b - 0.5, z\right) + t\right) + \color{blue}{a} \]

   if 5.50000000000000002e31 < y

   1. Initial program 98.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(a + \left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} + y \cdot i \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + t\right) + a\right) + y \cdot i \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + z\right) + t\right) + a\right) + y \cdot i \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 88.7

         \[\leadsto \left(\left(\mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, z\right) + t\right) + a\right) + y \cdot i \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right)} + y \cdot i \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right) + t\right) + a\right) + y \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 78.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z + t\right) + a\right) + y \cdot i \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
   10. Step-by-step derivation

   11. Applied rewrites 78.1%

       \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\log c, b - 0.5, z\right) + a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 \cdot z + t\right) + a\right) + y \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y 5.5e+31)
   (+ (fma (log c) (- b 0.5) z) a)
   (+ (+ (+ (* 1.0 z) t) a) (* y i))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= 5.5e+31) {
		tmp = fma(log(c), (b - 0.5), z) + a;
	} else {
		tmp = (((1.0 * z) + t) + a) + (y * i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= 5.5e+31)
		tmp = Float64(fma(log(c), Float64(b - 0.5), z) + a);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 * z) + t) + a) + Float64(y * i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, 5.5e+31], N[(N[(N[Log[c], $MachinePrecision] * N[(b - 0.5), $MachinePrecision] + z), $MachinePrecision] + a), $MachinePrecision], N[(N[(N[(N[(1.0 * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.50000000000000002e31

   1. Initial program 99.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 80.6

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 80.6%

      \[\leadsto \mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, \mathsf{fma}\left(i, y, z\right) + \left(a + t\right)\right) \]

   8. Taylor expanded in t around 0

      \[\leadsto a + \color{blue}{\left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 63.5%

       \[\leadsto \mathsf{fma}\left(i, y, \mathsf{fma}\left(\log c, b - 0.5, z\right)\right) + \color{blue}{a} \]

   11. Taylor expanded in y around 0

       \[\leadsto \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + a \]
   12. Step-by-step derivation

   13. Applied rewrites 60.5%

       \[\leadsto \mathsf{fma}\left(\log c, b - 0.5, z\right) + a \]

   if 5.50000000000000002e31 < y

   1. Initial program 98.9%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(a + \left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} + y \cdot i \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + t\right) + a\right) + y \cdot i \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + z\right) + t\right) + a\right) + y \cdot i \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 88.7

         \[\leadsto \left(\left(\mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, z\right) + t\right) + a\right) + y \cdot i \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right)} + y \cdot i \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right) + t\right) + a\right) + y \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 78.7%

      \[\leadsto \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z + t\right) + a\right) + y \cdot i \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
   10. Step-by-step derivation

   11. Applied rewrites 78.1%

       \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 70.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+211}:\\ \;\;\;\;\left(\left(1 \cdot z + t\right) + a\right) + y \cdot i\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= x 7.4e+211) (+ (+ (+ (* 1.0 z) t) a) (* y i)) (* (log y) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 7.4e+211) {
		tmp = (((1.0 * z) + t) + a) + (y * i);
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (x <= 7.4d+211) then
        tmp = (((1.0d0 * z) + t) + a) + (y * i)
    else
        tmp = log(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (x <= 7.4e+211) {
		tmp = (((1.0 * z) + t) + a) + (y * i);
	} else {
		tmp = Math.log(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if x <= 7.4e+211:
		tmp = (((1.0 * z) + t) + a) + (y * i)
	else:
		tmp = math.log(y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (x <= 7.4e+211)
		tmp = Float64(Float64(Float64(Float64(1.0 * z) + t) + a) + Float64(y * i));
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (x <= 7.4e+211)
		tmp = (((1.0 * z) + t) + a) + (y * i);
	else
		tmp = log(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[x, 7.4e+211], N[(N[(N[(N[(1.0 * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.40000000000000019e211

   1. Initial program 99.5%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(a + \left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} + y \cdot i \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

      3. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + t\right) + a\right) + y \cdot i \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + z\right) + t\right) + a\right) + y \cdot i \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 88.9

         \[\leadsto \left(\left(\mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, z\right) + t\right) + a\right) + y \cdot i \]

   5. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right)} + y \cdot i \]

   6. Taylor expanded in z around inf

      \[\leadsto \left(\left(z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right) + t\right) + a\right) + y \cdot i \]
   7. Step-by-step derivation

   8. Applied rewrites 78.1%

      \[\leadsto \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z + t\right) + a\right) + y \cdot i \]

   9. Taylor expanded in z around inf

      \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
   10. Step-by-step derivation

   11. Applied rewrites 71.3%

       \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]

   if 7.40000000000000019e211 < x

   1. Initial program 99.7%

      \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \left(t + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right) + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(a + t\right)} + \left(z + \left(i \cdot y + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) \]

      4. associate-+r+ N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\left(z + i \cdot y\right) + \log c \cdot \left(b - \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \left(a + t\right) + \color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + \left(z + i \cdot y\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(a + t\right) + \left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + \left(z + i \cdot y\right)\right) \]

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 38.5

          \[\leadsto \left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \color{blue}{\mathsf{fma}\left(i, y, z\right)}\right) \]

   5. Applied rewrites 38.5%

      \[\leadsto \color{blue}{\left(a + t\right) + \mathsf{fma}\left(b - 0.5, \log c, \mathsf{fma}\left(i, y, z\right)\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      3. lower-log.f64 63.9

         \[\leadsto \color{blue}{\log y} \cdot x \]

   8. Applied rewrites 63.9%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 68.4% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (+ (+ (+ (* 1.0 z) t) a) (* y i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((1.0 * z) + t) + a) + (y * i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = (((1.0d0 * z) + t) + a) + (y * i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return (((1.0 * z) + t) + a) + (y * i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return (((1.0 * z) + t) + a) + (y * i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(1.0 * z) + t) + a) + Float64(y * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = (((1.0 * z) + t) + a) + (y * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(1.0 * z), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + N[(y * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(a + \left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right)\right)} + y \cdot i \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(t + \left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right)\right) + a\right)} + y \cdot i \]

   3. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

   4. lower-+.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\left(z + \log c \cdot \left(b - \frac{1}{2}\right)\right) + t\right)} + a\right) + y \cdot i \]

   5. +-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\log c \cdot \left(b - \frac{1}{2}\right) + z\right)} + t\right) + a\right) + y \cdot i \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\left(\color{blue}{\left(b - \frac{1}{2}\right) \cdot \log c} + z\right) + t\right) + a\right) + y \cdot i \]

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 84.0

      \[\leadsto \left(\left(\mathsf{fma}\left(b - 0.5, \color{blue}{\log c}, z\right) + t\right) + a\right) + y \cdot i \]

5. Applied rewrites 84.0%

   \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(b - 0.5, \log c, z\right) + t\right) + a\right)} + y \cdot i \]

6. Taylor expanded in z around inf

   \[\leadsto \left(\left(z \cdot \left(1 + \frac{\log c \cdot \left(b - \frac{1}{2}\right)}{z}\right) + t\right) + a\right) + y \cdot i \]
7. Step-by-step derivation

8. Applied rewrites 73.8%

   \[\leadsto \left(\left(\mathsf{fma}\left(\log c, \frac{b - 0.5}{z}, 1\right) \cdot z + t\right) + a\right) + y \cdot i \]

9. Taylor expanded in z around inf

   \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
10. Step-by-step derivation

11. Applied rewrites 67.7%

    \[\leadsto \left(\left(1 \cdot z + t\right) + a\right) + y \cdot i \]
12. Add Preprocessing

Alternative 20: 23.5% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ i \cdot y \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (* i y))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return i * y;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = i * y
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return i * y;
}

Python

def code(x, y, z, t, a, b, c, i):
	return i * y

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(i * y)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = i * y;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(i * y), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(\left(\left(\left(x \cdot \log y + z\right) + t\right) + a\right) + \left(b - 0.5\right) \cdot \log c\right) + y \cdot i \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{i \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 21.6

      \[\leadsto \color{blue}{i \cdot y} \]

5. Applied rewrites 21.6%

   \[\leadsto \color{blue}{i \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024329 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))