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

Percentage Accurate: 93.6% → 98.6%

Time: 5.6s

Alternatives: 20

Speedup: 1.0×

• 31.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 93.6% accurate, 1.0× speedup

Math

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

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

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

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 93.6%

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

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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

3. Applied rewrites 97.8%

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

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 10^{+85}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), z, 0.083333333333333\right)}{x} - -0.91893853320467\right) - \mathsf{fma}\left(0.5 - x, \log x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+85)
   (-
    (-
     (/
      (fma
       (fma z (- y -0.0007936500793651) -0.0027777777777778)
       z
       0.083333333333333)
      x)
     -0.91893853320467)
    (fma (- 0.5 x) (log x) x))
   (fma
    (* y z)
    (/ z x)
    (-
     (/ 0.083333333333333 x)
     (fma (- 0.5 x) (log x) (- x 0.91893853320467))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+85) {
		tmp = ((fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), z, 0.083333333333333) / x) - -0.91893853320467) - fma((0.5 - x), log(x), x);
	} else {
		tmp = fma((y * z), (z / x), ((0.083333333333333 / x) - fma((0.5 - x), log(x), (x - 0.91893853320467))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+85)
		tmp = Float64(Float64(Float64(fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), z, 0.083333333333333) / x) - -0.91893853320467) - fma(Float64(0.5 - x), log(x), x));
	else
		tmp = fma(Float64(y * z), Float64(z / x), Float64(Float64(0.083333333333333 / x) - fma(Float64(0.5 - x), log(x), Float64(x - 0.91893853320467))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e+85], N[(N[(N[(N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - -0.91893853320467), $MachinePrecision] - N[(N[(0.5 - x), $MachinePrecision] * N[Log[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if x < 1e85

   1. Initial program 93.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) + \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} \]

      5. add-flip N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(\mathsf{neg}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(\mathsf{neg}\left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)}\right)\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \color{blue}{\left(x - \left(x - \frac{1}{2}\right) \cdot \log x\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(x - \left(x - \frac{1}{2}\right) \cdot \log x\right)} \]

   3. Applied rewrites 93.7%

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

   if 1e85 < x

   1. Initial program 93.6%

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

   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 85.0

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

   6. Applied rewrites 85.0%

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

Alternative 4: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, \log x, \left(0.91893853320467 - x\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0007936500793651 - y, z, 0.0027777777777778\right), z, -0.083333333333333\right)}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1e+85)
   (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)
     (fma (- 0.5 x) (log x) (- x 0.91893853320467))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1e+85) {
		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) - fma((0.5 - x), log(x), (x - 0.91893853320467))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1e+85)
		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) - fma(Float64(0.5 - x), log(x), Float64(x - 0.91893853320467))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1e+85], 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[(N[(0.5 - x), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(x - 0.91893853320467), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e85

   1. Initial program 93.6%

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

   3. Applied rewrites 93.7%

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

   if 1e85 < x

   1. Initial program 93.6%

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

   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 85.0

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

   6. Applied rewrites 85.0%

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

Alternative 5: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+64}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y \cdot z, z, 0.083333333333333\right)}{x} - -0.91893853320467\right) - \mathsf{fma}\left(0.5 - x, \log x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ (- y -0.0007936500793651) x) z) z)))
   (if (<= z -3.7e+84)
     t_0
     (if (<= z 7.5e+64)
       (-
        (- (/ (fma (* y z) z 0.083333333333333) x) -0.91893853320467)
        (fma (- 0.5 x) (log x) x))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double tmp;
	if (z <= -3.7e+84) {
		tmp = t_0;
	} else if (z <= 7.5e+64) {
		tmp = ((fma((y * z), z, 0.083333333333333) / x) - -0.91893853320467) - fma((0.5 - x), log(x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z)
	tmp = 0.0
	if (z <= -3.7e+84)
		tmp = t_0;
	elseif (z <= 7.5e+64)
		tmp = Float64(Float64(Float64(fma(Float64(y * z), z, 0.083333333333333) / x) - -0.91893853320467) - fma(Float64(0.5 - x), log(x), x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -3.7e+84], t$95$0, If[LessEqual[z, 7.5e+64], N[(N[(N[(N[(N[(y * z), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision] - -0.91893853320467), $MachinePrecision] - N[(N[(0.5 - x), $MachinePrecision] * N[Log[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7e84 or 7.5000000000000005e64 < z

   1. Initial program 93.6%

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

   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 42.1

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

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   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 44.0

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

      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 44.0

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

   6. Applied rewrites 44.0%

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

   if -3.7e84 < z < 7.5000000000000005e64

   1. Initial program 93.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) + \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} \]

      5. add-flip N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(\mathsf{neg}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(\mathsf{neg}\left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)}\right)\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \color{blue}{\left(x - \left(x - \frac{1}{2}\right) \cdot \log x\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(x - \left(x - \frac{1}{2}\right) \cdot \log x\right)} \]

   3. Applied rewrites 93.7%

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

   4. Taylor expanded in y around inf

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

      1. lower-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 6: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 350000:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \frac{z}{x}, \frac{0.083333333333333}{x} - \mathsf{fma}\left(0.5 - x, \log x, x - 0.91893853320467\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 350000.0)
   (/
    1.0
    (/
     x
     (fma
      (fma (- y -0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)))
   (fma
    (* y z)
    (/ z x)
    (-
     (/ 0.083333333333333 x)
     (fma (- 0.5 x) (log x) (- x 0.91893853320467))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 350000.0) {
		tmp = 1.0 / (x / fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333));
	} else {
		tmp = fma((y * z), (z / x), ((0.083333333333333 / x) - fma((0.5 - x), log(x), (x - 0.91893853320467))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 350000.0)
		tmp = Float64(1.0 / Float64(x / fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)));
	else
		tmp = fma(Float64(y * z), Float64(z / x), Float64(Float64(0.083333333333333 / x) - fma(Float64(0.5 - x), log(x), Float64(x - 0.91893853320467))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 350000.0], N[(1.0 / N[(x / N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5e5

   1. Initial program 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

         \[\leadsto \frac{1}{\frac{x}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{x}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{x}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{x}{\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 62.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}} \]

   if 3.5e5 < x

   1. Initial program 93.6%

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

   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 85.0

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

   6. Applied rewrites 85.0%

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

Alternative 7: 88.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x}\\ t_1 := \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\left(\frac{0.083333333333333}{x} - -0.91893853320467\right) - \mathsf{fma}\left(0.5 - x, \log x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467)
          (/
           (+
            (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
            0.083333333333333)
           x)))
        (t_1 (* (* (/ (- y -0.0007936500793651) x) z) z)))
   (if (<= t_0 -5e+45)
     t_1
     (if (<= t_0 2e+304)
       (-
        (- (/ 0.083333333333333 x) -0.91893853320467)
        (fma (- 0.5 x) (log x) x))
       t_1))))

C

double code(double x, double y, double z) {
	double t_0 = ((((x - 0.5) * log(x)) - x) + 0.91893853320467) + ((((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333) / x);
	double t_1 = (((y - -0.0007936500793651) / x) * z) * z;
	double tmp;
	if (t_0 <= -5e+45) {
		tmp = t_1;
	} else if (t_0 <= 2e+304) {
		tmp = ((0.083333333333333 / x) - -0.91893853320467) - fma((0.5 - x), log(x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = 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))
	t_1 = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z)
	tmp = 0.0
	if (t_0 <= -5e+45)
		tmp = t_1;
	elseif (t_0 <= 2e+304)
		tmp = Float64(Float64(Float64(0.083333333333333 / x) - -0.91893853320467) - fma(Float64(0.5 - x), log(x), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+45], t$95$1, If[LessEqual[t$95$0, 2e+304], N[(N[(N[(0.083333333333333 / x), $MachinePrecision] - -0.91893853320467), $MachinePrecision] - N[(N[(0.5 - x), $MachinePrecision] * N[Log[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < -5e45 or 1.9999999999999999e304 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x))

   1. Initial program 93.6%

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

   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 42.1

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

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   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 44.0

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

      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 44.0

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

   6. Applied rewrites 44.0%

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

   if -5e45 < (+.f64 (+.f64 (-.f64 (*.f64 (-.f64 x #s(literal 1/2 binary64)) (log.f64 x)) x) #s(literal 91893853320467/100000000000000 binary64)) (/.f64 (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) x)) < 1.9999999999999999e304

   1. Initial program 93.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) + \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} \]

      5. add-flip N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(\mathsf{neg}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(\mathsf{neg}\left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)}\right)\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \color{blue}{\left(x - \left(x - \frac{1}{2}\right) \cdot \log x\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(x - \left(x - \frac{1}{2}\right) \cdot \log x\right)} \]

   3. Applied rewrites 93.7%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 57.3%

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

Alternative 8: 83.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, y - -0.0007936500793651, -0.0027777777777778\right), \frac{z}{x}, \frac{0.083333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.05e+85)
   (fma
    (fma z (- y -0.0007936500793651) -0.0027777777777778)
    (/ z x)
    (/ 0.083333333333333 x))
   (* x (- (* -1.0 (log (/ 1.0 x))) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.05e+85) {
		tmp = fma(fma(z, (y - -0.0007936500793651), -0.0027777777777778), (z / x), (0.083333333333333 / x));
	} else {
		tmp = x * ((-1.0 * log((1.0 / x))) - 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.05e+85)
		tmp = fma(fma(z, Float64(y - -0.0007936500793651), -0.0027777777777778), Float64(z / x), Float64(0.083333333333333 / x));
	else
		tmp = Float64(x * Float64(Float64(-1.0 * log(Float64(1.0 / x))) - 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.05e+85], N[(N[(z * N[(y - -0.0007936500793651), $MachinePrecision] + -0.0027777777777778), $MachinePrecision] * N[(z / x), $MachinePrecision] + N[(0.083333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(-1.0 * N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000005e85

   1. Initial program 93.6%

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

   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 65.1

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

   6. Applied rewrites 65.1%

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

   if 1.05000000000000005e85 < x

   1. Initial program 93.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      4. lower-log.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      5. lower-/.f64 35.1

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

   4. Applied rewrites 35.1%

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

Alternative 9: 83.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\frac{1}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.05e+85)
   (/
    1.0
    (/
     x
     (fma
      (fma (- y -0.0007936500793651) z -0.0027777777777778)
      z
      0.083333333333333)))
   (* x (- (* -1.0 (log (/ 1.0 x))) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.05e+85) {
		tmp = 1.0 / (x / fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333));
	} else {
		tmp = x * ((-1.0 * log((1.0 / x))) - 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.05e+85)
		tmp = Float64(1.0 / Float64(x / fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333)));
	else
		tmp = Float64(x * Float64(Float64(-1.0 * log(Float64(1.0 / x))) - 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.05e+85], N[(1.0 / N[(x / N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(-1.0 * N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000005e85

   1. Initial program 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-flip N/A

         \[\leadsto \frac{1}{\frac{x}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{1}{\frac{x}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{1}{\frac{x}{\left(z \cdot \left(\frac{7936500793651}{10000000000000000} + y\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\frac{x}{\left(z \cdot \left(y + \frac{7936500793651}{10000000000000000}\right) + \left(\mathsf{neg}\left(\frac{13888888888889}{5000000000000000}\right)\right)\right) \cdot z + \frac{83333333333333}{1000000000000000}}} \]

      12. add-flip N/A

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

      13. metadata-eval N/A

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

      14. lift--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lower-unsound-/.f64 N/A

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

   6. Applied rewrites 62.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}}} \]

   if 1.05000000000000005e85 < x

   1. Initial program 93.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      4. lower-log.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      5. lower-/.f64 35.1

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

   4. Applied rewrites 35.1%

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

Alternative 10: 83.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0.91893853320467 - \mathsf{fma}\left(0.5 - x, \log x, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.05e+85)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (- 0.91893853320467 (fma (- 0.5 x) (log x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.05e+85) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = 0.91893853320467 - fma((0.5 - x), log(x), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.05e+85)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(0.91893853320467 - fma(Float64(0.5 - x), log(x), x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.05e+85], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(0.91893853320467 - N[(N[(0.5 - x), $MachinePrecision] * N[Log[x], $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000005e85

   1. Initial program 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   6. Applied rewrites 62.8%

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

   if 1.05000000000000005e85 < x

   1. Initial program 93.6%

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467\right) + \frac{\left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333}{x} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right) + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}} \]

      2. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \frac{91893853320467}{100000000000000}\right)} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x} \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right) + \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) + \left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)} \]

      5. add-flip N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(\mathsf{neg}\left(\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(\mathsf{neg}\left(\color{blue}{\left(\left(x - \frac{1}{2}\right) \cdot \log x - x\right)}\right)\right) \]

      7. sub-negate-rev N/A

         \[\leadsto \left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \color{blue}{\left(x - \left(x - \frac{1}{2}\right) \cdot \log x\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{91893853320467}{100000000000000} + \frac{\left(\left(y + \frac{7936500793651}{10000000000000000}\right) \cdot z - \frac{13888888888889}{5000000000000000}\right) \cdot z + \frac{83333333333333}{1000000000000000}}{x}\right) - \left(x - \left(x - \frac{1}{2}\right) \cdot \log x\right)} \]

   3. Applied rewrites 93.7%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.5%

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

Alternative 11: 83.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.05e+85)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* x (- (* -1.0 (log (/ 1.0 x))) 1.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.05e+85) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = x * ((-1.0 * log((1.0 / x))) - 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.05e+85)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x);
	else
		tmp = Float64(x * Float64(Float64(-1.0 * log(Float64(1.0 / x))) - 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.05e+85], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $MachinePrecision] * z + 0.083333333333333), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(-1.0 * N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000005e85

   1. Initial program 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   6. Applied rewrites 62.8%

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

   if 1.05000000000000005e85 < x

   1. Initial program 93.6%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      4. lower-log.f64 N/A

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

      5. lower-/.f64 35.1

         \[\leadsto x \cdot \left(-1 \cdot \log \left(\frac{1}{x}\right) - 1\right) \]

   4. Applied rewrites 35.1%

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

Alternative 12: 65.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y - -0.0007936500793651, z, -0.0027777777777778\right), z, 0.083333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.05e+71)
   (/
    (fma
     (fma (- y -0.0007936500793651) z -0.0027777777777778)
     z
     0.083333333333333)
    x)
   (* (* (/ (- y -0.0007936500793651) x) z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.05e+71) {
		tmp = fma(fma((y - -0.0007936500793651), z, -0.0027777777777778), z, 0.083333333333333) / x;
	} else {
		tmp = (((y - -0.0007936500793651) / x) * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.05e+71)
		tmp = Float64(fma(fma(Float64(y - -0.0007936500793651), z, -0.0027777777777778), 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[x, 1.05e+71], N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] * z + -0.0027777777777778), $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 x < 1.04999999999999995e71

   1. Initial program 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   6. Applied rewrites 62.8%

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

   if 1.04999999999999995e71 < x

   1. Initial program 93.6%

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

   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 42.1

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

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   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 44.0

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

      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 44.0

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

   6. Applied rewrites 44.0%

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

Alternative 13: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\frac{0.083333333333333 + z \cdot \left(0.0007936500793651 \cdot z - 0.0027777777777778\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ (- y -0.0007936500793651) x) z) z))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+24)
     t_0
     (if (<= t_1 5e+32)
       (/
        (+
         0.083333333333333
         (* z (- (* 0.0007936500793651 z) 0.0027777777777778)))
        x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = t_0;
	} else if (t_1 <= 5e+32) {
		tmp = (0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((y - (-0.0007936500793651d0)) / x) * z) * z
    t_1 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_1 <= (-1d+24)) then
        tmp = t_0
    else if (t_1 <= 5d+32) then
        tmp = (0.083333333333333d0 + (z * ((0.0007936500793651d0 * z) - 0.0027777777777778d0))) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = t_0;
	} else if (t_1 <= 5e+32) {
		tmp = (0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y - -0.0007936500793651) / x) * z) * z
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_1 <= -1e+24:
		tmp = t_0
	elif t_1 <= 5e+32:
		tmp = (0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+24)
		tmp = t_0;
	elseif (t_1 <= 5e+32)
		tmp = Float64(Float64(0.083333333333333 + Float64(z * Float64(Float64(0.0007936500793651 * z) - 0.0027777777777778))) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_1 <= -1e+24)
		tmp = t_0;
	elseif (t_1 <= 5e+32)
		tmp = (0.083333333333333 + (z * ((0.0007936500793651 * z) - 0.0027777777777778))) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $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+24], t$95$0, If[LessEqual[t$95$1, 5e+32], N[(N[(0.083333333333333 + N[(z * N[(N[(0.0007936500793651 * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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)) < -9.9999999999999998e23 or 4.9999999999999997e32 < (+.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 93.6%

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

   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 42.1

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

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   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 44.0

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

      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 44.0

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

   6. Applied rewrites 44.0%

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

   if -9.9999999999999998e23 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 4.9999999999999997e32

   1. Initial program 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 46.4

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

   7. Applied rewrites 46.4%

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

Alternative 14: 64.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{y - -0.0007936500793651}{x} \cdot z\right) \cdot z\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.1:\\ \;\;\;\;\frac{1}{\frac{x}{0.083333333333333}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* (/ (- y -0.0007936500793651) x) z) z))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+24)
     t_0
     (if (<= t_1 0.1) (/ 1.0 (/ x 0.083333333333333)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = t_0;
	} else if (t_1 <= 0.1) {
		tmp = 1.0 / (x / 0.083333333333333);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((y - (-0.0007936500793651d0)) / x) * z) * z
    t_1 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_1 <= (-1d+24)) then
        tmp = t_0
    else if (t_1 <= 0.1d0) then
        tmp = 1.0d0 / (x / 0.083333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = t_0;
	} else if (t_1 <= 0.1) {
		tmp = 1.0 / (x / 0.083333333333333);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y - -0.0007936500793651) / x) * z) * z
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_1 <= -1e+24:
		tmp = t_0
	elif t_1 <= 0.1:
		tmp = 1.0 / (x / 0.083333333333333)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y - -0.0007936500793651) / x) * z) * z)
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+24)
		tmp = t_0;
	elseif (t_1 <= 0.1)
		tmp = Float64(1.0 / Float64(x / 0.083333333333333));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y - -0.0007936500793651) / x) * z) * z;
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_1 <= -1e+24)
		tmp = t_0;
	elseif (t_1 <= 0.1)
		tmp = 1.0 / (x / 0.083333333333333);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y - -0.0007936500793651), $MachinePrecision] / x), $MachinePrecision] * z), $MachinePrecision] * z), $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+24], t$95$0, If[LessEqual[t$95$1, 0.1], N[(1.0 / N[(x / 0.083333333333333), $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)) < -9.9999999999999998e23 or 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 93.6%

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

   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 42.1

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

   4. Applied rewrites 42.1%

      \[\leadsto \color{blue}{{z}^{2} \cdot \mathsf{fma}\left(0.0007936500793651, \frac{1}{x}, \frac{y}{x}\right)} \]
   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 44.0

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

      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 44.0

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

   6. Applied rewrites 44.0%

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

   if -9.9999999999999998e23 < (+.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 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 23.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 23.3

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

   9. Applied rewrites 23.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{0.083333333333333}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+24}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{y}{x}\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\frac{1}{\frac{x}{0.083333333333333}}\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot \frac{0.0007936500793651}{x}\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_0 -1e+24)
     (* (* z z) (/ y x))
     (if (<= t_0 500.0)
       (/ 1.0 (/ x 0.083333333333333))
       (* (* z z) (/ 0.0007936500793651 x))))))

C

double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -1e+24) {
		tmp = (z * z) * (y / x);
	} else if (t_0 <= 500.0) {
		tmp = 1.0 / (x / 0.083333333333333);
	} else {
		tmp = (z * z) * (0.0007936500793651 / x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_0 <= (-1d+24)) then
        tmp = (z * z) * (y / x)
    else if (t_0 <= 500.0d0) then
        tmp = 1.0d0 / (x / 0.083333333333333d0)
    else
        tmp = (z * z) * (0.0007936500793651d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_0 <= -1e+24) {
		tmp = (z * z) * (y / x);
	} else if (t_0 <= 500.0) {
		tmp = 1.0 / (x / 0.083333333333333);
	} else {
		tmp = (z * z) * (0.0007936500793651 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_0 <= -1e+24:
		tmp = (z * z) * (y / x)
	elif t_0 <= 500.0:
		tmp = 1.0 / (x / 0.083333333333333)
	else:
		tmp = (z * z) * (0.0007936500793651 / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_0 <= -1e+24)
		tmp = Float64(Float64(z * z) * Float64(y / x));
	elseif (t_0 <= 500.0)
		tmp = Float64(1.0 / Float64(x / 0.083333333333333));
	else
		tmp = Float64(Float64(z * z) * Float64(0.0007936500793651 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_0 <= -1e+24)
		tmp = (z * z) * (y / x);
	elseif (t_0 <= 500.0)
		tmp = 1.0 / (x / 0.083333333333333);
	else
		tmp = (z * z) * (0.0007936500793651 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(y + 0.0007936500793651), $MachinePrecision] * z), $MachinePrecision] - 0.0027777777777778), $MachinePrecision] * z), $MachinePrecision] + 0.083333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+24], N[(N[(z * z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(1.0 / N[(x / 0.083333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * N[(0.0007936500793651 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < -9.9999999999999998e23

   1. Initial program 93.6%

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

   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 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 42.1

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

   6. Applied rewrites 42.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 42.1

          \[\leadsto \left(z \cdot z\right) \cdot \left(\frac{1}{x} \cdot \color{blue}{\left(y - -0.0007936500793651\right)}\right) \]

   8. Applied rewrites 42.1%

      \[\leadsto \left(z \cdot z\right) \cdot \left(\frac{1}{x} \cdot \color{blue}{\left(y - -0.0007936500793651\right)}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(z \cdot z\right) \cdot \frac{y}{\color{blue}{x}} \]
   10. Step-by-step derivation

       1. lower-/.f64 30.4

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

   11. Applied rewrites 30.4%

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

   if -9.9999999999999998e23 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 500

   1. Initial program 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 23.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 23.3

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

   9. Applied rewrites 23.3%

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

   if 500 < (+.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 93.6%

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

   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 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 42.1

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

   6. Applied rewrites 42.1%

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

   7. Taylor expanded in y around 0

      \[\leadsto \left(z \cdot z\right) \cdot \frac{\frac{7936500793651}{10000000000000000}}{\color{blue}{x}} \]
   8. Step-by-step derivation

      1. lower-/.f64 25.7

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

   9. Applied rewrites 25.7%

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

Alternative 16: 50.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \frac{y}{x}\\ t_1 := \left(\left(y + 0.0007936500793651\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.083333333333333\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+68}:\\ \;\;\;\;\frac{1}{\frac{x}{0.083333333333333}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z z) (/ y x)))
        (t_1
         (+
          (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)
          0.083333333333333)))
   (if (<= t_1 -1e+24)
     t_0
     (if (<= t_1 4e+68) (/ 1.0 (/ x 0.083333333333333)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z * z) * (y / x);
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = t_0;
	} else if (t_1 <= 4e+68) {
		tmp = 1.0 / (x / 0.083333333333333);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (z * z) * (y / x)
    t_1 = ((((y + 0.0007936500793651d0) * z) - 0.0027777777777778d0) * z) + 0.083333333333333d0
    if (t_1 <= (-1d+24)) then
        tmp = t_0
    else if (t_1 <= 4d+68) then
        tmp = 1.0d0 / (x / 0.083333333333333d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * z) * (y / x);
	double t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	double tmp;
	if (t_1 <= -1e+24) {
		tmp = t_0;
	} else if (t_1 <= 4e+68) {
		tmp = 1.0 / (x / 0.083333333333333);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * z) * (y / x)
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333
	tmp = 0
	if t_1 <= -1e+24:
		tmp = t_0
	elif t_1 <= 4e+68:
		tmp = 1.0 / (x / 0.083333333333333)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * z) * Float64(y / x))
	t_1 = Float64(Float64(Float64(Float64(Float64(y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333)
	tmp = 0.0
	if (t_1 <= -1e+24)
		tmp = t_0;
	elseif (t_1 <= 4e+68)
		tmp = Float64(1.0 / Float64(x / 0.083333333333333));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * z) * (y / x);
	t_1 = ((((y + 0.0007936500793651) * z) - 0.0027777777777778) * z) + 0.083333333333333;
	tmp = 0.0;
	if (t_1 <= -1e+24)
		tmp = t_0;
	elseif (t_1 <= 4e+68)
		tmp = 1.0 / (x / 0.083333333333333);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(y / 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+24], t$95$0, If[LessEqual[t$95$1, 4e+68], N[(1.0 / N[(x / 0.083333333333333), $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)) < -9.9999999999999998e23 or 3.99999999999999981e68 < (+.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 93.6%

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

   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 42.1

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

   4. Applied rewrites 42.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 42.1

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

   6. Applied rewrites 42.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 42.1

          \[\leadsto \left(z \cdot z\right) \cdot \left(\frac{1}{x} \cdot \color{blue}{\left(y - -0.0007936500793651\right)}\right) \]

   8. Applied rewrites 42.1%

      \[\leadsto \left(z \cdot z\right) \cdot \left(\frac{1}{x} \cdot \color{blue}{\left(y - -0.0007936500793651\right)}\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto \left(z \cdot z\right) \cdot \frac{y}{\color{blue}{x}} \]
   10. Step-by-step derivation

       1. lower-/.f64 30.4

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

   11. Applied rewrites 30.4%

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

   if -9.9999999999999998e23 < (+.f64 (*.f64 (-.f64 (*.f64 (+.f64 y #s(literal 7936500793651/10000000000000000 binary64)) z) #s(literal 13888888888889/5000000000000000 binary64)) z) #s(literal 83333333333333/1000000000000000 binary64)) < 3.99999999999999981e68

   1. Initial program 93.6%

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

   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 62.8

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 23.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 23.3

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

   9. Applied rewrites 23.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{0.083333333333333}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 28.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

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 62.8

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in z around 0

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

   1. lower-*.f64 28.7

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

7. Applied rewrites 28.7%

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

Alternative 18: 23.3% accurate, 4.9× speedup

Math

\[\frac{1}{\frac{x}{0.083333333333333}} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 (/ x 0.083333333333333)))

C

double code(double x, double y, double z) {
	return 1.0 / (x / 0.083333333333333);
}

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 = 1.0d0 / (x / 0.083333333333333d0)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / (x / 0.083333333333333);
}

Python

def code(x, y, z):
	return 1.0 / (x / 0.083333333333333)

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(x / 0.083333333333333))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / (x / 0.083333333333333);
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(x / 0.083333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

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 62.8

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 23.3%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 23.3

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

9. Applied rewrites 23.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{x}{0.083333333333333}}} \]
10. Add Preprocessing

Alternative 19: 23.3% accurate, 5.2× speedup

Math

\[\frac{1}{x} \cdot 0.083333333333333 \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 x) 0.083333333333333))

C

double code(double x, double y, double z) {
	return (1.0 / x) * 0.083333333333333;
}

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 = (1.0d0 / x) * 0.083333333333333d0
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / x) * 0.083333333333333;
}

Python

def code(x, y, z):
	return (1.0 / x) * 0.083333333333333

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / x) * 0.083333333333333)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / x) * 0.083333333333333;
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / x), $MachinePrecision] * 0.083333333333333), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

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 62.8

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 23.3%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lift-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 23.3

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

9. Applied rewrites 23.3%

   \[\leadsto \frac{1}{x} \cdot \color{blue}{0.083333333333333} \]
10. Add Preprocessing

Alternative 20: 23.3% accurate, 8.7× speedup

Math

\[\frac{0.083333333333333}{x} \]

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 93.6%

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

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 62.8

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

4. Applied rewrites 62.8%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 23.3%

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

Reproduce

herbie shell --seed 2025172 
(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)))