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

Percentage Accurate: 94.0% → 98.6%

Time: 5.8s

Alternatives: 16

Speedup: 1.0×

• 31.5% 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: 94.0% 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: 98.6% 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 94.0%

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

3. Applied rewrites 98.6%

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

Alternative 2: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x} - x \cdot \left(1 + \log \left(\frac{1}{x}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8e+189)
   (fma
    (- x 0.5)
    (log x)
    (-
     (- 0.91893853320467 x)
     (/
      (fma
       (fma (- -0.0007936500793651 y) z 0.0027777777777778)
       z
       -0.083333333333333)
      x)))
   (fma
    (* y z)
    (/ z x)
    (- (/ 0.083333333333333 x) (* x (+ 1.0 (log (/ 1.0 x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8e+189) {
		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((y * z), (z / x), ((0.083333333333333 / x) - (x * (1.0 + log((1.0 / x))))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8e+189)
		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(Float64(y * z), Float64(z / x), Float64(Float64(0.083333333333333 / x) - Float64(x * Float64(1.0 + log(Float64(1.0 / x))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8e+189], 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[(N[(y * z), $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]]

Derivation

1. Split input into 2 regimes

2. if x < 8.0000000000000002e189

   1. Initial program 94.0%

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

   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 8.0000000000000002e189 < x

   1. Initial program 94.0%

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 83.0

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

   9. Applied rewrites 83.0%

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

Alternative 3: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right)\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          (* y z)
          (/ z x)
          (-
           (/ 0.083333333333333 x)
           (fma (- 0.5 x) (log x) (- x 0.91893853320467))))))
   (if (<= y -2.65e+26)
     t_0
     (if (<= y 5e-93)
       (fma
        (- x 0.5)
        (log x)
        (-
         (- 0.91893853320467 x)
         (/
          (fma
           (fma -0.0007936500793651 z 0.0027777777777778)
           z
           -0.083333333333333)
          x)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y * z), (z / x), ((0.083333333333333 / x) - fma((0.5 - x), log(x), (x - 0.91893853320467))));
	double tmp;
	if (y <= -2.65e+26) {
		tmp = t_0;
	} else if (y <= 5e-93) {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) - (fma(fma(-0.0007936500793651, z, 0.0027777777777778), z, -0.083333333333333) / x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y * z), Float64(z / x), Float64(Float64(0.083333333333333 / x) - fma(Float64(0.5 - x), log(x), Float64(x - 0.91893853320467))))
	tmp = 0.0
	if (y <= -2.65e+26)
		tmp = t_0;
	elseif (y <= 5e-93)
		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 = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * z), $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]}, If[LessEqual[y, -2.65e+26], t$95$0, If[LessEqual[y, 5e-93], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.64999999999999984e26 or 4.99999999999999994e-93 < y

   1. Initial program 94.0%

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

   if -2.64999999999999984e26 < y < 4.99999999999999994e-93

   1. Initial program 94.0%

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

   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%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-0.0007936500793651}, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x} - x \cdot \left(1 + \log \left(\frac{1}{x}\right)\right)\right)\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          (* y z)
          (/ z x)
          (- (/ 0.083333333333333 x) (* x (+ 1.0 (log (/ 1.0 x))))))))
   (if (<= y -2.65e+26)
     t_0
     (if (<= y 4.2e-80)
       (fma
        (- x 0.5)
        (log x)
        (-
         (- 0.91893853320467 x)
         (/
          (fma
           (fma -0.0007936500793651 z 0.0027777777777778)
           z
           -0.083333333333333)
          x)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((y * z), (z / x), ((0.083333333333333 / x) - (x * (1.0 + log((1.0 / x))))));
	double tmp;
	if (y <= -2.65e+26) {
		tmp = t_0;
	} else if (y <= 4.2e-80) {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) - (fma(fma(-0.0007936500793651, z, 0.0027777777777778), z, -0.083333333333333) / x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(y * z), Float64(z / x), Float64(Float64(0.083333333333333 / x) - Float64(x * Float64(1.0 + log(Float64(1.0 / x))))))
	tmp = 0.0
	if (y <= -2.65e+26)
		tmp = t_0;
	elseif (y <= 4.2e-80)
		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 = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * z), $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]}, If[LessEqual[y, -2.65e+26], t$95$0, If[LessEqual[y, 4.2e-80], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.64999999999999984e26 or 4.20000000000000003e-80 < y

   1. Initial program 94.0%

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 83.0

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

   9. Applied rewrites 83.0%

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

   if -2.64999999999999984e26 < y < 4.20000000000000003e-80

   1. Initial program 94.0%

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

   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%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-0.0007936500793651}, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} - \mathsf{fma}\left(\log x, 0.5 - x, x - 0.91893853320467\right)\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (-
          (/ (fma (* y z) z 0.083333333333333) x)
          (fma (log x) (- 0.5 x) (- x 0.91893853320467)))))
   (if (<= y -2.65e+26)
     t_0
     (if (<= y 2e-94)
       (fma
        (- x 0.5)
        (log x)
        (-
         (- 0.91893853320467 x)
         (/
          (fma
           (fma -0.0007936500793651 z 0.0027777777777778)
           z
           -0.083333333333333)
          x)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (fma((y * z), z, 0.083333333333333) / x) - fma(log(x), (0.5 - x), (x - 0.91893853320467));
	double tmp;
	if (y <= -2.65e+26) {
		tmp = t_0;
	} else if (y <= 2e-94) {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) - (fma(fma(-0.0007936500793651, z, 0.0027777777777778), z, -0.083333333333333) / x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(fma(Float64(y * z), z, 0.083333333333333) / x) - fma(log(x), Float64(0.5 - x), Float64(x - 0.91893853320467)))
	tmp = 0.0
	if (y <= -2.65e+26)
		tmp = t_0;
	elseif (y <= 2e-94)
		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 = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] * N[(0.5 - x), $MachinePrecision] + N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.65e+26], t$95$0, If[LessEqual[y, 2e-94], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.64999999999999984e26 or 1.9999999999999999e-94 < y

   1. Initial program 94.0%

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

   8. Applied rewrites 81.8%

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

   if -2.64999999999999984e26 < y < 1.9999999999999999e-94

   1. Initial program 94.0%

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

   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%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{-0.0007936500793651}, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+260}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} - \mathsf{fma}\left(\log x, 0.5 - x, x - 0.91893853320467\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.8e-15)
   (fma
    (fma z (- y -0.0007936500793651) -0.0027777777777778)
    (/ z x)
    (/ 0.083333333333333 x))
   (if (<= x 1.75e+260)
     (-
      (/ (fma (* y z) z 0.083333333333333) x)
      (fma (log x) (- 0.5 x) (- x 0.91893853320467)))
     (fma (- x 0.5) (log x) (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.8e-15) {
		tmp = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	} else if (x <= 1.75e+260) {
		tmp = (fma((y * z), z, 0.083333333333333) / x) - fma(log(x), (0.5 - x), (x - 0.91893853320467));
	} else {
		tmp = fma((x - 0.5), log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.8e-15)
		tmp = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x));
	elseif (x <= 1.75e+260)
		tmp = Float64(Float64(fma(Float64(y * z), z, 0.083333333333333) / x) - fma(log(x), Float64(0.5 - x), Float64(x - 0.91893853320467)));
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.8e-15], N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75e+260], N[(N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - N[(N[Log[x], $MachinePrecision] * N[(0.5 - x), $MachinePrecision] + N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.8000000000000001e-15

   1. Initial program 94.0%

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

   3. Applied rewrites 98.6%

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

   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 66.0

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

   6. Applied rewrites 66.0%

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

   if 6.8000000000000001e-15 < x < 1.7499999999999999e260

   1. Initial program 94.0%

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 83.9

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

   6. Applied rewrites 83.9%

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

   8. Applied rewrites 81.8%

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

   if 1.7499999999999999e260 < x

   1. Initial program 94.0%

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

   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 35.6

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

   6. Applied rewrites 35.6%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, -1 \cdot \color{blue}{x}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \mathsf{neg}\left(x\right)\right) \]

      3. lower-neg.f64 35.6

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

   8. Applied rewrites 35.6%

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

Alternative 7: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{-0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          (fma z (- y -0.0007936500793651) -0.0027777777777778)
          (/ z x)
          (/ 0.083333333333333 x)))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+115)
     t_0
     (if (<= t_1 5e+83)
       (fma
        (- x 0.5)
        (log x)
        (- (- 0.91893853320467 x) (/ -0.083333333333333 x)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+115) {
		tmp = t_0;
	} else if (t_1 <= 5e+83) {
		tmp = fma((x - 0.5), log(x), ((0.91893853320467 - x) - (-0.083333333333333 / x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x))
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+115)
		tmp = t_0;
	elseif (t_1 <= 5e+83)
		tmp = fma(Float64(x - 0.5), log(x), Float64(Float64(0.91893853320467 - x) - Float64(-0.083333333333333 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+115], t$95$0, If[LessEqual[t$95$1, 5e+83], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[(0.91893853320467 - x), $MachinePrecision] - N[(-0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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)) < -1e115 or 5.00000000000000029e83 < (+.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 94.0%

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

   3. Applied rewrites 98.6%

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

   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 66.0

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

   6. Applied rewrites 66.0%

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

   if -1e115 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 5.00000000000000029e83

   1. Initial program 94.0%

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

   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 56.9%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\color{blue}{-0.083333333333333}}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\frac{0.083333333333333}{x} - \left(x - \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (fma
          (fma z (- y -0.0007936500793651) -0.0027777777777778)
          (/ z x)
          (/ 0.083333333333333 x)))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+115)
     t_0
     (if (<= t_1 5e+83)
       (-
        (/ 0.083333333333333 x)
        (- x (fma (log x) (- x 0.5) 0.91893853320467)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+115) {
		tmp = t_0;
	} else if (t_1 <= 5e+83) {
		tmp = (0.083333333333333 / x) - (x - fma(log(x), (x - 0.5), 0.91893853320467));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x))
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+115)
		tmp = t_0;
	elseif (t_1 <= 5e+83)
		tmp = Float64(Float64(0.083333333333333 / x) - Float64(x - fma(log(x), Float64(x - 0.5), 0.91893853320467)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+115], t$95$0, If[LessEqual[t$95$1, 5e+83], N[(N[(0.083333333333333 / x), $MachinePrecision] - N[(x - N[(N[Log[x], $MachinePrecision] * N[(x - 0.5), $MachinePrecision] + 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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)) < -1e115 or 5.00000000000000029e83 < (+.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 94.0%

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

   3. Applied rewrites 98.6%

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

   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 66.0

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

   6. Applied rewrites 66.0%

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

   if -1e115 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 5.00000000000000029e83

   1. Initial program 94.0%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 56.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+l- N/A

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

      8. sub-negate-rev N/A

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

      9. associate--l- N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-log.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lower--.f64 N/A

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

   6. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{0.083333333333333}{x} - \left(x - \mathsf{fma}\left(\log x, x - 0.5, 0.91893853320467\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7000000.0)
   (fma
    (fma z (- y -0.0007936500793651) -0.0027777777777778)
    (/ z x)
    (/ 0.083333333333333 x))
   (fma (- x 0.5) (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7000000.0) {
		tmp = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	} else {
		tmp = fma((x - 0.5), log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7000000.0)
		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(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7000000.0], 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] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e6

   1. Initial program 94.0%

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

   3. Applied rewrites 98.6%

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

   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 66.0

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

   6. Applied rewrites 66.0%

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

   if 7e6 < x

   1. Initial program 94.0%

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

   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 35.6

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

   6. Applied rewrites 35.6%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, -1 \cdot \color{blue}{x}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \mathsf{neg}\left(x\right)\right) \]

      3. lower-neg.f64 35.6

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

   8. Applied rewrites 35.6%

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

Alternative 10: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7000000.0)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (fma (- x 0.5) (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7000000.0) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = fma((x - 0.5), log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7000000.0)
		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(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7000000.0], 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] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e6

   1. Initial program 94.0%

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

   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 63.8

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

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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 63.8

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 63.8

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

   6. Applied rewrites 63.8%

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

   if 7e6 < x

   1. Initial program 94.0%

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

   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 35.6

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

   6. Applied rewrites 35.6%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, -1 \cdot \color{blue}{x}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \mathsf{neg}\left(x\right)\right) \]

      3. lower-neg.f64 35.6

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

   8. Applied rewrites 35.6%

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

Alternative 11: 84.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot \left(0.0007936500793651 + y\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7000000.0)
   (/ (fma (* z (+ 0.0007936500793651 y)) z 0.083333333333333) x)
   (fma (- x 0.5) (log x) (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7000000.0) {
		tmp = fma((z * (0.0007936500793651 + y)), z, 0.083333333333333) / x;
	} else {
		tmp = fma((x - 0.5), log(x), -x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7000000.0)
		tmp = Float64(fma(Float64(z * Float64(0.0007936500793651 + y)), z, 0.083333333333333) / x);
	else
		tmp = fma(Float64(x - 0.5), log(x), Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7000000.0], N[(N[(N[(z * N[(0.0007936500793651 + y), $MachinePrecision]), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * N[Log[x], $MachinePrecision] + (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e6

   1. Initial program 94.0%

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

   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 63.8

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

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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 63.8

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 63.8

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 63.4

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

   9. Applied rewrites 63.4%

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

   if 7e6 < x

   1. Initial program 94.0%

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

   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 35.6

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

   6. Applied rewrites 35.6%

      \[\leadsto \mathsf{fma}\left(x - 0.5, \log x, \color{blue}{-1 \cdot x}\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, -1 \cdot \color{blue}{x}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, \log x, \mathsf{neg}\left(x\right)\right) \]

      3. lower-neg.f64 35.6

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

   8. Applied rewrites 35.6%

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

Alternative 12: 64.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333 \leq 0.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
       0.083333333333333)
      0.1)
   (/ (fma (* y z) z 0.083333333333333) x)
   (* (* (/ (- y -0.0007936500793651) x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) <= 0.1) {
		tmp = fma((y * z), z, 0.083333333333333) / x;
	} else {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	}
	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) <= 0.1)
		tmp = Float64(fma(Float64(y * z), z, 0.083333333333333) / x);
	else
		tmp = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z);
	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], 0.1], N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $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)) < 0.10000000000000001

   1. Initial program 94.0%

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

   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 63.8

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

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
   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 63.8

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 63.8

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 50.7

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

   9. Applied rewrites 50.7%

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

   if 0.10000000000000001 < (+.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 94.0%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{{z}^{2} \cdot \left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\left(\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x} + \frac{y}{x}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {z}^{2} \cdot \left(\color{blue}{\frac{7936500793651}{10000000000000000} \cdot \frac{1}{x}} + \frac{y}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \color{blue}{\frac{1}{x}}, \frac{y}{x}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{\color{blue}{x}}, \frac{y}{x}\right) \]

      5. lower-/.f64 42.6

         \[\leadsto {z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \]

   4. Applied rewrites 42.6%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {z}^{2} \cdot \color{blue}{\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \color{blue}{{z}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot {z}^{\color{blue}{2}} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot \left(z \cdot \color{blue}{z}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{7936500793651}{10000000000000000}, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot \color{blue}{z} \]

      7. lower-*.f64 44.5

         \[\leadsto \left(\mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right) \cdot z\right) \cdot z \]

      8. lift-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. div-add-rev N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. lower-/.f64 44.5

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

   6. Applied rewrites 44.5%

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

Alternative 13: 63.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

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 63.8

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

4. Applied rewrites 63.8%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
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 63.8

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 63.8

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

6. Applied rewrites 63.8%

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

7. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 63.4

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

9. Applied rewrites 63.4%

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

Alternative 14: 50.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

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 63.8

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

4. Applied rewrites 63.8%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
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 63.8

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 63.8

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

6. Applied rewrites 63.8%

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

7. Taylor expanded in y around inf

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

   1. lower-*.f64 50.7

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

9. Applied rewrites 50.7%

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

Alternative 15: 29.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

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 63.8

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

4. Applied rewrites 63.8%

   \[\leadsto \color{blue}{\frac{0.083333333333333 + z \cdot \left(z \cdot \left(0.0007936500793651 + y\right) - 0.0027777777777778\right)}{x}} \]
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 63.8

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

   16. lift-fma.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f64 63.8

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

6. Applied rewrites 63.8%

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

7. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{-13888888888889}{5000000000000000}, z, \frac{83333333333333}{1000000000000000}\right)}{x} \]
8. Step-by-step derivation

9. Applied rewrites 29.7%

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

Alternative 16: 23.8% 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 94.0%

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

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 63.8

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

4. Applied rewrites 63.8%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 23.8%

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

Reproduce

herbie shell --seed 2025143 
(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)))