Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Percentage Accurate: 94.1% → 98.8%

Time: 7.3s

Alternatives: 16

Speedup: 1.0×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
  (/
   (+
    (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
    0.083333333333333)
   x)))

C

double code(double x, double y, double z) {
	return ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((((x - 0.5d0) * log(x)) - x) + 0.91893853320467d0) + ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) / x)
end function

Java

public static double code(double x, double y, double z) {
	return ((((x - 0.5) * Math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
}

Python

def code(x, y, z):
	return ((((x - 0.5) * math.log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(x - 0.5) * log(x)) - x) + 0.91893853320467) + Float64(Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] + 0.91893853320467), $MachinePrecision] + N[(N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma
  (fma z (- y -0.0007936500793651) -0.0027777777777778)
  (/ z x)
  (- (/ 0.083333333333333 x) (fma (- 0.5 x) (log x) (- x 0.91893853320467)))))

C

double code(double x, double y, double z) {
	return fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), ((0.083333333333333 / x) - fma((0.5 - x), log(x), (x - 0.91893853320467))));
}

Julia

function code(x, y, z)
	return fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(Float64(0.083333333333333 / x) - fma(Float64(0.5 - x), log(x), Float64(x - 0.91893853320467))))
end

Wolfram

code[x_, y_, z_] := N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - N[(N[(0.5 - x), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

3. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right)} \]
4. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma
  (/ (fma z (- y -0.0007936500793651) -0.0027777777777778) x)
  z
  (- (/ 0.083333333333333 x) (fma (- 0.5 x) (log x) (- x 0.91893853320467)))))

C

double code(double x, double y, double z) {
	return fma((fma(z, (y - -0.0007936500793651), -0.0027777777777778) / x), z, ((0.083333333333333 / x) - fma((0.5 - x), log(x), (x - 0.91893853320467))));
}

Julia

function code(x, y, z)
	return fma(Float64(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778) / x), z, Float64(Float64(0.083333333333333 / x) - fma(Float64(0.5 - x), log(x), Float64(x - 0.91893853320467))))
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] * z + N[(N[(0.083333333333333 / x), $MachinePrecision] - N[(N[(0.5 - x), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

3. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right)}{x}, z, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right)} \]
4. Add Preprocessing

Alternative 3: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, y, 0.0007936500793651 \cdot z\right)}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      1e+299)
   (fma
    (- x 0.5)
    (log x)
    (-
     (- 0.91893853320467 x)
     (/
      (fma
       (fma (- -0.0007936500793651 y) z 0.0027777777777778)
       z
       -0.083333333333333)
      x)))
   (fma z (/ (fma z y (* 0.0007936500793651 z)) x) (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+299) {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) - (fma(fma((-0.0007936500793651 - y), z, 0.0027777777777778), z, -0.083333333333333) / x)));
	} else {
		tmp = fma(z, (fma(z, y, (0.0007936500793651 * z)) / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+299)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(0.91893853320467 - x) - Float64(fma(fma(Float64(-0.0007936500793651 - y), z, 0.0027777777777778), z, -0.083333333333333) / x)));
	else
		tmp = fma(z, Float64(fma(z, y, Float64(0.0007936500793651 * z)) / x), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 1e+299], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 - x), $MachinePrecision] - N[(N[(N[(N[(-0.0007936500793651 - y), $MachinePrecision] * z + 0.0027777777777778), $MachinePrecision] * z + -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(z * y + N[(0.0007936500793651 * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1.0000000000000001e299

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)} \]

   if 1.0000000000000001e299 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   6. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}}, \frac{0.083333333333333}{x}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      2. lower-+.f64 64.1

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]

   9. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

       2. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

       4. distribute-lft-in N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot y + z \cdot \frac{7936500793651}{10000000000000000}}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, y, z \cdot \frac{7936500793651}{10000000000000000}\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, y, \frac{7936500793651}{10000000000000000} \cdot z\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

       7. lower-*.f64 64.1

          \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, y, 0.0007936500793651 \cdot z\right)}{x}, \frac{0.083333333333333}{x}\right) \]

   11. Applied rewrites 64.1%

       \[\leadsto \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, y, 0.0007936500793651 \cdot z\right)}{x}, \frac{0.083333333333333}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) - \left(x - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{if}\;y \leq -0.00078:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} - x \cdot \left(1 + \log \left(\frac{1}{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (-
          (fma (- x 0.5) (log x) 0.91893853320467)
          (- x (/ (fma (* y z) z 0.083333333333333) x)))))
   (if (<= y -0.00078)
     t_0
     (if (<= y 1.2e-92)
       (fma
        (fma z 0.0007936500793651 -0.0027777777777778)
        (/ z x)
        (- (/ 0.083333333333333 x) (* x (+ 1.0 (log (/ 1.0 x))))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x - 0.5), log(x), 0.91893853320467) - (x - (fma((y * z), z, 0.083333333333333) / x));
	double tmp;
	if (y <= -0.00078) {
		tmp = t_0;
	} else if (y <= 1.2e-92) {
		tmp = fma(fma(z, 0.0007936500793651, -0.0027777777777778), (z / x), ((0.083333333333333 / x) - (x * (1.0 + log((1.0 / x))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) - Float64(x - Float64(fma(Float64(y * z), z, 0.083333333333333) / x)))
	tmp = 0.0
	if (y <= -0.00078)
		tmp = t_0;
	elseif (y <= 1.2e-92)
		tmp = fma(fma(z, 0.0007936500793651, -0.0027777777777778), Float64(z / x), Float64(Float64(0.083333333333333 / x) - Float64(x * Float64(1.0 + log(Float64(1.0 / x))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(x - N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.00078], t$95$0, If[LessEqual[y, 1.2e-92], N[(N[(z * 0.0007936500793651 + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(N[(0.083333333333333 / x), $MachinePrecision] - N[(x * N[(1.0 + N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.79999999999999986e-4 or 1.2000000000000001e-92 < y

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in y around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(y \cdot z\right)} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   3. Step-by-step derivation

      1. lower-*.f64 82.4

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(y \cdot \color{blue}{z}\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied rewrites 82.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(y \cdot z\right)} \cdot z + 0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(x - \frac{91893853320467}{100000000000000}\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(x - \frac{91893853320467}{100000000000000}\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. sub-negate-rev N/A

         \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{91893853320467}{100000000000000} - x\right)\right)\right)}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      8. lift--.f64 N/A

         \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)}\right)\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      9. add-flip N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{91893853320467}{100000000000000} - x\right)\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      10. lift--.f64 N/A

          \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      11. associate-+r- N/A

          \[\leadsto \color{blue}{\left(\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - x\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      12. associate-+l- N/A

          \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - \left(x - \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - \left(x - \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

   6. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) - \left(x - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)} \]

   if -7.79999999999999986e-4 < y < 1.2000000000000001e-92

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{\frac{7936500793651}{10000000000000000}}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} - \mathsf{fma}\left(\frac{1}{2} - x, \log x, x - \frac{91893853320467}{100000000000000}\right)\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 80.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \color{blue}{0.0007936500793651}, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right) \]

   7. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} - \color{blue}{x \cdot \left(1 + \log \left(\frac{1}{x}\right)\right)}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x \cdot \color{blue}{\left(1 + \log \left(\frac{1}{x}\right)\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x \cdot \left(1 + \color{blue}{\log \left(\frac{1}{x}\right)}\right)\right) \]

      3. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, \frac{7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x} - x \cdot \left(1 + \log \left(\frac{1}{x}\right)\right)\right) \]

      4. lower-/.f64 79.7

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} - x \cdot \left(1 + \log \left(\frac{1}{x}\right)\right)\right) \]

   9. Applied rewrites 79.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} - \color{blue}{x \cdot \left(1 + \log \left(\frac{1}{x}\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) - \left(x - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+180}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -1e+69)
     (-
      (fma (- x 0.5) (log x) 0.91893853320467)
      (- x (/ (fma (* y z) z 0.083333333333333) x)))
     (if (<= t_0 1e+180)
       (fma
        (- x 0.5)
        (log x)
        (-
         (- 0.91893853320467 x)
         (/
          (fma
           (fma -0.0007936500793651 z 0.0027777777777778)
           z
           -0.083333333333333)
          x)))
       (fma
        (* (- y -0.0007936500793651) z)
        (/ z x)
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -1e+69) {
		tmp = fma((x - 0.5), log(x), 0.91893853320467) - (x - (fma((y * z), z, 0.083333333333333) / x));
	} else if (t_0 <= 1e+180) {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) - (fma(fma(-0.0007936500793651, z, 0.0027777777777778), z, -0.083333333333333) / x)));
	} else {
		tmp = fma(((y - -0.0007936500793651) * z), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -1e+69)
		tmp = Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) - Float64(x - Float64(fma(Float64(y * z), z, 0.083333333333333) / x)));
	elseif (t_0 <= 1e+180)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(0.91893853320467 - x) - Float64(fma(fma(-0.0007936500793651, z, 0.0027777777777778), z, -0.083333333333333) / x)));
	else
		tmp = fma(Float64(Float64(y - -0.0007936500793651) * z), Float64(z / x), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+69], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(x - N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+180], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 - x), $MachinePrecision] - N[(N[(N[(-0.0007936500793651 * z + 0.0027777777777778), $MachinePrecision] * z + -0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -1.0000000000000001e69

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in y around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(y \cdot z\right)} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   3. Step-by-step derivation

      1. lower-*.f64 82.4

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(y \cdot \color{blue}{z}\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied rewrites 82.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(y \cdot z\right)} \cdot z + 0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(x - \frac{91893853320467}{100000000000000}\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(x - \frac{91893853320467}{100000000000000}\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. sub-negate-rev N/A

         \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{91893853320467}{100000000000000} - x\right)\right)\right)}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      8. lift--.f64 N/A

         \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)}\right)\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      9. add-flip N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{91893853320467}{100000000000000} - x\right)\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      10. lift--.f64 N/A

          \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      11. associate-+r- N/A

          \[\leadsto \color{blue}{\left(\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - x\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      12. associate-+l- N/A

          \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - \left(x - \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - \left(x - \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

   6. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) - \left(x - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)} \]

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 1e180

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)} \]

   4. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-7936500793651}{10000000000000000}}, z, \frac{13888888888889}{5000000000000000}\right), z, \frac{-83333333333333}{1000000000000000}\right)}{x}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 78.6%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-0.0007936500793651}, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right) \]

   if 1e180 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   6. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}}, \frac{0.083333333333333}{x}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      2. lower-+.f64 64.1

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]

   9. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]

       2. *-commutative N/A

          \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \cdot z + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       4. mult-flip N/A

          \[\leadsto \left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{1}{x}\right) \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       5. lift-/.f64 N/A

          \[\leadsto \left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{1}{x}\right) \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       6. associate-*l* N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(\frac{1}{x} \cdot z\right) + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       7. *-commutative N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(z \cdot \frac{1}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       8. lift-/.f64 N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(z \cdot \frac{1}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       9. mult-flip N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       10. lift-/.f64 N/A

           \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       11. lower-fma.f64 64.9

           \[\leadsto \mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right), \color{blue}{\frac{z}{x}}, \frac{0.083333333333333}{x}\right) \]

   11. Applied rewrites 64.9%

       \[\leadsto \mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, \color{blue}{\frac{z}{x}}, \frac{0.083333333333333}{x}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) - \left(x - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      1e+31)
   (-
    (fma (- x 0.5) (log x) 0.91893853320467)
    (- x (/ (fma (* y z) z 0.083333333333333) x)))
   (fma (* (- y -0.0007936500793651) z) (/ z x) (/ 0.083333333333333 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+31) {
		tmp = fma((x - 0.5), log(x), 0.91893853320467) - (x - (fma((y * z), z, 0.083333333333333) / x));
	} else {
		tmp = fma(((y - -0.0007936500793651) * z), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 1e+31)
		tmp = Float64(fma(Float64(x - 0.5), log(x), 0.91893853320467) - Float64(x - Float64(fma(Float64(y * z), z, 0.083333333333333) / x)));
	else
		tmp = fma(Float64(Float64(y - -0.0007936500793651) * z), Float64(z / x), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 1e+31], N[(N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + 0.91893853320467), $MachinePrecision] - N[(x - N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 9.9999999999999996e30

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in y around inf

      \[\leadsto \left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\color{blue}{\left(y \cdot z\right)} \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]
   3. Step-by-step derivation

      1. lower-*.f64 82.4

         \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(y \cdot \color{blue}{z}\right) \cdot z + 0.083333333333333}{x} \]

   4. Applied rewrites 82.4%

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\color{blue}{\left(y \cdot z\right)} \cdot z + 0.083333333333333}{x} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift--.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} + \frac{91893853320467}{100000000000000}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      4. associate-+l- N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - \left(x - \frac{91893853320467}{100000000000000}\right)\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(x - \frac{1}{2}\right) \cdot \log x} - \left(x - \frac{91893853320467}{100000000000000}\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{\log x \cdot \left(x - \frac{1}{2}\right)} - \left(x - \frac{91893853320467}{100000000000000}\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. sub-negate-rev N/A

         \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{91893853320467}{100000000000000} - x\right)\right)\right)}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      8. lift--.f64 N/A

         \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)}\right)\right)\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      9. add-flip N/A

         \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \left(\frac{91893853320467}{100000000000000} - x\right)\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      10. lift--.f64 N/A

          \[\leadsto \left(\log x \cdot \left(x - \frac{1}{2}\right) + \color{blue}{\left(\frac{91893853320467}{100000000000000} - x\right)}\right) + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      11. associate-+r- N/A

          \[\leadsto \color{blue}{\left(\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - x\right)} + \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      12. associate-+l- N/A

          \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - \left(x - \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      13. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\log x \cdot \left(x - \frac{1}{2}\right) + \frac{91893853320467}{100000000000000}\right) - \left(x - \frac{\left(y \cdot z\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

   6. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, 0.91893853320467\right) - \left(x - \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\right)} \]

   if 9.9999999999999996e30 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   6. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}}, \frac{0.083333333333333}{x}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      2. lower-+.f64 64.1

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]

   9. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]

       2. *-commutative N/A

          \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \cdot z + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       4. mult-flip N/A

          \[\leadsto \left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{1}{x}\right) \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       5. lift-/.f64 N/A

          \[\leadsto \left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{1}{x}\right) \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       6. associate-*l* N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(\frac{1}{x} \cdot z\right) + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       7. *-commutative N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(z \cdot \frac{1}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       8. lift-/.f64 N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(z \cdot \frac{1}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       9. mult-flip N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       10. lift-/.f64 N/A

           \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       11. lower-fma.f64 64.9

           \[\leadsto \mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right), \color{blue}{\frac{z}{x}}, \frac{0.083333333333333}{x}\right) \]

   11. Applied rewrites 64.9%

       \[\leadsto \mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, \color{blue}{\frac{z}{x}}, \frac{0.083333333333333}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y \cdot z}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{-0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -1e+69)
     (fma z (/ (* y z) x) (/ 0.083333333333333 x))
     (if (<= t_0 1e+31)
       (fma
        (- x 0.5)
        (log x)
        (- (- 0.91893853320467 x) (/ -0.083333333333333 x)))
       (fma
        (* (- y -0.0007936500793651) z)
        (/ z x)
        (/ 0.083333333333333 x))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -1e+69) {
		tmp = fma(z, ((y * z) / x), (0.083333333333333 / x));
	} else if (t_0 <= 1e+31) {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) - (-0.083333333333333 / x)));
	} else {
		tmp = fma(((y - -0.0007936500793651) * z), (z / x), (0.083333333333333 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -1e+69)
		tmp = fma(z, Float64(Float64(y * z) / x), Float64(0.083333333333333 / x));
	elseif (t_0 <= 1e+31)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(0.91893853320467 - x) - Float64(-0.083333333333333 / x)));
	else
		tmp = fma(Float64(Float64(y - -0.0007936500793651) * z), Float64(z / x), Float64(0.083333333333333 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+69], N[(z * N[(N[(y * z), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+31], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 - x), $MachinePrecision] - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -1.0000000000000001e69

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   6. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}}, \frac{0.083333333333333}{x}\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{y \cdot z}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 50.0

         \[\leadsto \mathsf{fma}\left(z, \frac{y \cdot z}{x}, \frac{0.083333333333333}{x}\right) \]

   9. Applied rewrites 50.0%

      \[\leadsto \mathsf{fma}\left(z, \frac{y \cdot z}{x}, \frac{0.083333333333333}{x}\right) \]

   if -1.0000000000000001e69 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 9.9999999999999996e30

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \left(\frac{91893853320467}{100000000000000} - x\right) - \frac{\color{blue}{\frac{-83333333333333}{1000000000000000}}}{x}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 57.6%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\color{blue}{-0.083333333333333}}{x}\right) \]

   if 9.9999999999999996e30 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   6. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}}, \frac{0.083333333333333}{x}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      2. lower-+.f64 64.1

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]

   9. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]

       2. *-commutative N/A

          \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \cdot z + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       4. mult-flip N/A

          \[\leadsto \left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{1}{x}\right) \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       5. lift-/.f64 N/A

          \[\leadsto \left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{1}{x}\right) \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       6. associate-*l* N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(\frac{1}{x} \cdot z\right) + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       7. *-commutative N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(z \cdot \frac{1}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       8. lift-/.f64 N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(z \cdot \frac{1}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       9. mult-flip N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       10. lift-/.f64 N/A

           \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       11. lower-fma.f64 64.9

           \[\leadsto \mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right), \color{blue}{\frac{z}{x}}, \frac{0.083333333333333}{x}\right) \]

   11. Applied rewrites 64.9%

       \[\leadsto \mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, \color{blue}{\frac{z}{x}}, \frac{0.083333333333333}{x}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 83.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.9e+99)
   (fma
    (fma z (- y -0.0007936500793651) -0.0027777777777778)
    (/ z x)
    (/ 0.083333333333333 x))
   (fma (- x 0.5) (log x) (* -1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.9e+99) {
		tmp = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	} else {
		tmp = fma((x - 0.5), log(x), (-1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.9e+99)
		tmp = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(-1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.9e+99], N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9e99

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right), \frac{z}{x}, \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 65.3

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{\color{blue}{x}}\right) \]

   6. Applied rewrites 65.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \color{blue}{\frac{0.083333333333333}{x}}\right) \]

   if 1.9e99 < x

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{-1 \cdot x}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 36.3

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

   6. Applied rewrites 36.3%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 83.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.9e+99)
   (fma
    z
    (/ (fma (- y -0.0007936500793651) z -0.0027777777777778) x)
    (/ 0.083333333333333 x))
   (fma (- x 0.5) (log x) (* -1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.9e+99) {
		tmp = fma(z, (fma((y - -0.0007936500793651), z, -0.0027777777777778) / x), (0.083333333333333 / x));
	} else {
		tmp = fma((x - 0.5), log(x), (-1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.9e+99)
		tmp = fma(z, Float64(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778) / x), Float64(0.083333333333333 / x));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(-1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.9e+99], N[(z * N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9e99

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   6. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}}, \frac{0.083333333333333}{x}\right) \]

   if 1.9e99 < x

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{-1 \cdot x}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 36.3

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

   6. Applied rewrites 36.3%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 83.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.9e+99)
   (fma (* (- y -0.0007936500793651) z) (/ z x) (/ 0.083333333333333 x))
   (fma (- x 0.5) (log x) (* -1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.9e+99) {
		tmp = fma(((y - -0.0007936500793651) * z), (z / x), (0.083333333333333 / x));
	} else {
		tmp = fma((x - 0.5), log(x), (-1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.9e+99)
		tmp = fma(Float64(Float64(y - -0.0007936500793651) * z), Float64(z / x), Float64(0.083333333333333 / x));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(-1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.9e+99], N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9e99

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   6. Applied rewrites 64.5%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}}, \frac{0.083333333333333}{x}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

      2. lower-+.f64 64.1

         \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]

   9. Applied rewrites 64.1%

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]

       2. *-commutative N/A

          \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \cdot z + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       4. mult-flip N/A

          \[\leadsto \left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{1}{x}\right) \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       5. lift-/.f64 N/A

          \[\leadsto \left(\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{1}{x}\right) \cdot z + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       6. associate-*l* N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(\frac{1}{x} \cdot z\right) + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       7. *-commutative N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(z \cdot \frac{1}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       8. lift-/.f64 N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \left(z \cdot \frac{1}{x}\right) + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       9. mult-flip N/A

          \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       10. lift-/.f64 N/A

           \[\leadsto \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) \cdot \frac{z}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

       11. lower-fma.f64 64.9

           \[\leadsto \mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right), \color{blue}{\frac{z}{x}}, \frac{0.083333333333333}{x}\right) \]

   11. Applied rewrites 64.9%

       \[\leadsto \mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, \color{blue}{\frac{z}{x}}, \frac{0.083333333333333}{x}\right) \]

   if 1.9e99 < x

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{-1 \cdot x}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 36.3

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

   6. Applied rewrites 36.3%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 82.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -1 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.7e+99)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (fma (- x 0.5) (log x) (* -1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.7e+99) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = fma((x - 0.5), log(x), (-1.0 * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.7e+99)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(-1.0 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.7e+99], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999992e99

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      2. +-commutative N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      8. add-flip N/A

         \[\leadsto \frac{\left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \left(\mathsf{neg}\left(\frac{-13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      13. add-flip N/A

          \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      15. lift-fma.f64 63.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   6. Applied rewrites 63.3%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

   if 1.69999999999999992e99 < x

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   3. Applied rewrites 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \color{blue}{-1 \cdot x}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 36.3

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

   6. Applied rewrites 36.3%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (fma
   (fma (- y -0.0007936500793651) z -0.0027777777777778)
   z
   0.083333333333333)
  x))

C

double code(double x, double y, double z) {
	return fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.3

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.3%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   2. +-commutative N/A

      \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) + \frac{83333333333333}{1000000000000000}}{x} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   5. lift--.f64 N/A

      \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   7. +-commutative N/A

      \[\leadsto \frac{\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   8. add-flip N/A

      \[\leadsto \frac{\left(z \cdot \left(y - \left(\mathsf{neg}\left(\frac{7936500793651}{10000000000000000}\right)\right)\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   10. lift--.f64 N/A

       \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) - \left(\mathsf{neg}\left(\frac{-13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   13. add-flip N/A

       \[\leadsto \frac{\left(z \cdot \left(y - \frac{-7936500793651}{10000000000000000}\right) + \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   14. lift-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(z, y - \frac{-7936500793651}{10000000000000000}, \frac{-13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

   15. lift-fma.f64 63.3

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]

6. Applied rewrites 63.3%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} \]
7. Add Preprocessing

Alternative 13: 62.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, z, 0.083333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (fma (* (- y -0.0007936500793651) z) z 0.083333333333333) x))

C

double code(double x, double y, double z) {
	return fma(((y - -0.0007936500793651) * z), z, 0.083333333333333) / x;
}

Julia

function code(x, y, z)
	return Float64(fma(Float64(Float64(y - -0.0007936500793651) * z), z, 0.083333333333333) / x)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.3

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.3%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

6. Applied rewrites 64.5%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right)}{x}}, \frac{0.083333333333333}{x}\right) \]

7. Taylor expanded in z around inf

   \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x}, \frac{\frac{83333333333333}{1000000000000000}}{x}\right) \]

   2. lower-+.f64 64.1

      \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]

9. Applied rewrites 64.1%

   \[\leadsto \mathsf{fma}\left(z, \frac{z \cdot \left(0.0007936500793651 + y\right)}{x}, \frac{0.083333333333333}{x}\right) \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} + \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]

    2. lift-/.f64 N/A

       \[\leadsto z \cdot \frac{z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{x} \]

    3. associate-*r/ N/A

       \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)}{x} + \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

    4. lift-/.f64 N/A

       \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right)}{x} + \frac{\frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]

    5. div-add-rev N/A

       \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]

    6. lower-/.f64 N/A

       \[\leadsto \frac{z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right)\right) + \frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]

11. Applied rewrites 62.8%

    \[\leadsto \frac{\mathsf{fma}\left(\left(y - -0.0007936500793651\right) \cdot z, z, 0.083333333333333\right)}{\color{blue}{x}} \]
12. Add Preprocessing

Alternative 14: 33.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 500000000000:\\ \;\;\;\;\frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.083333333333333}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      500000000000.0)
   (/ (+ 0.083333333333333 (* -0.0027777777777778 z)) x)
   (/ (* x 0.083333333333333) (* x x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000000.0) {
		tmp = (0.083333333333333 + (-0.0027777777777778 * z)) / x;
	} else {
		tmp = (x * 0.083333333333333) / (x * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0) <= 500000000000.0d0) then
        tmp = (0.083333333333333d0 + ((-0.0027777777777778d0) * z)) / x
    else
        tmp = (x * 0.083333333333333d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000000.0) {
		tmp = (0.083333333333333 + (-0.0027777777777778 * z)) / x;
	} else {
		tmp = (x * 0.083333333333333) / (x * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000000.0:
		tmp = (0.083333333333333 + (-0.0027777777777778 * z)) / x
	else:
		tmp = (x * 0.083333333333333) / (x * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000000.0)
		tmp = Float64(Float64(0.083333333333333 + Float64(-0.0027777777777778 * z)) / x);
	else
		tmp = Float64(Float64(x * 0.083333333333333) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 500000000000.0)
		tmp = (0.083333333333333 + (-0.0027777777777778 * z)) / x;
	else
		tmp = (x * 0.083333333333333) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision], 500000000000.0], N[(N[(0.083333333333333 + N[(-0.0027777777777778 * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x * 0.083333333333333), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 5e11

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{x} \]
   6. Step-by-step derivation

      1. lower-*.f64 29.3

         \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]

   7. Applied rewrites 29.3%

      \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]

   if 5e11 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64))

   1. Initial program 94.1%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

      6. lower-+.f64 63.3

         \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

   4. Applied rewrites 63.3%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.7%

      \[\leadsto \frac{0.083333333333333}{x} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]

      2. mult-flip N/A

         \[\leadsto \frac{83333333333333}{1000000000000000} \cdot \color{blue}{\frac{1}{x}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{83333333333333}{1000000000000000} \cdot \frac{1}{\color{blue}{x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

      5. lower-*.f64 23.6

         \[\leadsto \frac{1}{x} \cdot \color{blue}{0.083333333333333} \]

   9. Applied rewrites 23.6%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{0.083333333333333} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{83333333333333}{1000000000000000}} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{1}{x} \cdot \frac{83333333333333}{1000000000000000} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{1 \cdot \frac{83333333333333}{1000000000000000}}{\color{blue}{x}} \]

       4. associate-/l* N/A

          \[\leadsto 1 \cdot \color{blue}{\frac{\frac{83333333333333}{1000000000000000}}{x}} \]

       5. *-inverses N/A

          \[\leadsto \frac{x}{x} \cdot \frac{\color{blue}{\frac{83333333333333}{1000000000000000}}}{x} \]

       6. frac-times N/A

          \[\leadsto \frac{x \cdot \frac{83333333333333}{1000000000000000}}{\color{blue}{x \cdot x}} \]

       7. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot \frac{83333333333333}{1000000000000000}}{x \cdot \color{blue}{x}} \]

       8. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot \frac{83333333333333}{1000000000000000}}{\color{blue}{x \cdot x}} \]

       9. lower-*.f64 21.9

          \[\leadsto \frac{x \cdot 0.083333333333333}{\color{blue}{x} \cdot x} \]

   11. Applied rewrites 21.9%

       \[\leadsto \frac{x \cdot 0.083333333333333}{\color{blue}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 29.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (+ 0.083333333333333 (* -0.0027777777777778 z)) x))

C

double code(double x, double y, double z) {
	return (0.083333333333333 + (-0.0027777777777778 * z)) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (0.083333333333333d0 + ((-0.0027777777777778d0) * z)) / x
end function

Java

public static double code(double x, double y, double z) {
	return (0.083333333333333 + (-0.0027777777777778 * z)) / x;
}

Python

def code(x, y, z):
	return (0.083333333333333 + (-0.0027777777777778 * z)) / x

Julia

function code(x, y, z)
	return Float64(Float64(0.083333333333333 + Float64(-0.0027777777777778 * z)) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (0.083333333333333 + (-0.0027777777777778 * z)) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(0.083333333333333 + N[(-0.0027777777777778 * z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.3

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.3%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + \frac{-13888888888889}{5000000000000000} \cdot z}{x} \]
6. Step-by-step derivation

   1. lower-*.f64 29.3

      \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]

7. Applied rewrites 29.3%

   \[\leadsto \frac{0.083333333333333 + -0.0027777777777778 \cdot z}{x} \]
8. Add Preprocessing

Alternative 16: 23.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.083333333333333}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 0.083333333333333 x))

C

double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.083333333333333d0 / x
end function

Java

public static double code(double x, double y, double z) {
	return 0.083333333333333 / x;
}

Python

def code(x, y, z):
	return 0.083333333333333 / x

Julia

function code(x, y, z)
	return Float64(0.083333333333333 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.083333333333333 / x;
end

Wolfram

code[x_, y_, z_] := N[(0.083333333333333 / x), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{\color{blue}{x}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{83333333333333}{1000000000000000} + z \cdot \left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) - \frac{13888888888889}{5000000000000000}\right)}{x} \]

   6. lower-+.f64 63.3

      \[\leadsto \frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x} \]

4. Applied rewrites 63.3%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\frac{83333333333333}{1000000000000000}}{x} \]
6. Step-by-step derivation

7. Applied rewrites 23.7%

   \[\leadsto \frac{0.083333333333333}{x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))