bug500, discussion (missed optimization)

Percentage Accurate: 53.0% → 97.9%

Time: 7.2s

Alternatives: 9

Speedup: 19.3×

• could not determine a ground truth

  1. x = -3.3533549280067656e+249

Specification

Math

\[\begin{array}{l} \\ \log \left(\frac{\sinh x}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ (sinh x) x)))

C

double code(double x) {
	return log((sinh(x) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = log((sinh(x) / x))
end function

Java

public static double code(double x) {
	return Math.log((Math.sinh(x) / x));
}

Python

def code(x):
	return math.log((math.sinh(x) / x))

Julia

function code(x)
	return log(Float64(sinh(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = log((sinh(x) / x));
end

Wolfram

code[x_] := N[Log[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\sinh x}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ (sinh x) x)))

C

double code(double x) {
	return log((sinh(x) / 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = log((sinh(x) / x))
end function

Java

public static double code(double x) {
	return Math.log((Math.sinh(x) / x));
}

Python

def code(x):
	return math.log((math.sinh(x) / x))

Julia

function code(x)
	return log(Float64(sinh(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = log((sinh(x) / x));
end

Wolfram

code[x_] := N[Log[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.086:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right) \cdot x\_m, x\_m, -0.005555555555555556\right) \cdot x\_m, x\_m, 0.16666666666666666\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{-1}{x\_m} \cdot \sinh \left(-x\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.086)
   (*
    (fma
     (*
      (fma
       (* (fma (* x_m x_m) -2.6455026455026456e-5 0.0003527336860670194) x_m)
       x_m
       -0.005555555555555556)
      x_m)
     x_m
     0.16666666666666666)
    (* x_m x_m))
   (log (* (/ -1.0 x_m) (sinh (- x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.086) {
		tmp = fma((fma((fma((x_m * x_m), -2.6455026455026456e-5, 0.0003527336860670194) * x_m), x_m, -0.005555555555555556) * x_m), x_m, 0.16666666666666666) * (x_m * x_m);
	} else {
		tmp = log(((-1.0 / x_m) * sinh(-x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.086)
		tmp = Float64(fma(Float64(fma(Float64(fma(Float64(x_m * x_m), -2.6455026455026456e-5, 0.0003527336860670194) * x_m), x_m, -0.005555555555555556) * x_m), x_m, 0.16666666666666666) * Float64(x_m * x_m));
	else
		tmp = log(Float64(Float64(-1.0 / x_m) * sinh(Float64(-x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.086], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.6455026455026456e-5 + 0.0003527336860670194), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + -0.005555555555555556), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(-1.0 / x$95$m), $MachinePrecision] * N[Sinh[(-x$95$m)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.085999999999999993

   1. Initial program 53.4%

      \[\log \left(\frac{\sinh x}{x}\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.6455026455026456 \cdot 10^{-5}, x \cdot x, 0.0003527336860670194\right) \cdot x\right) \cdot x - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot \left(x \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{37800}, x \cdot x, \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{37800}, x \cdot x, \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{37800}, x \cdot x, \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}\right) \cdot {x}^{2} + \frac{1}{6}\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

      9. pow2 N/A

         \[\leadsto \left(\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}\right) \cdot x\right) \cdot x + \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}\right) \cdot x, x, \frac{1}{6}\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

   6. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right) \cdot x, x, -0.005555555555555556\right) \cdot x, x, 0.16666666666666666\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

   if 0.085999999999999993 < x

   1. Initial program 42.4%

      \[\log \left(\frac{\sinh x}{x}\right) \]

   2. Taylor expanded in x around -inf

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

      1. sum-log N/A

         \[\leadsto \log \left(\left(\frac{-1}{2} \cdot \left(e^{x} - \frac{1}{e^{x}}\right)\right) \cdot \frac{-1}{x}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto \log \left(\left(\frac{-1}{2} \cdot \left(e^{x} - \frac{1}{e^{x}}\right)\right) \cdot \frac{-1}{x}\right) \]

      3. *-commutative N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(e^{x} - \frac{1}{e^{x}}\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(e^{x} - \frac{1}{e^{x}}\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(e^{x} - \frac{1}{e^{x}}\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(e^{x} - \frac{1}{e^{x}}\right)\right)\right) \]

      7. rec-exp N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(e^{x} - e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. sinh-undef N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(2 \cdot \sinh x\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(2 \cdot \sinh x\right)\right)\right) \]

      10. lift-sinh.f64 42.4

          \[\leadsto \log \left(\frac{-1}{x} \cdot \left(-0.5 \cdot \left(2 \cdot \sinh x\right)\right)\right) \]

   4. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\log \left(\frac{-1}{x} \cdot \left(-0.5 \cdot \left(2 \cdot \sinh x\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(2 \cdot \sinh x\right)\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(2 \cdot \sinh x\right)\right)\right) \]

      3. lift-sinh.f64 N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\frac{-1}{2} \cdot \left(2 \cdot \sinh x\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \log \left(\frac{-1}{x} \cdot \left(\left(\frac{-1}{2} \cdot 2\right) \cdot \sinh x\right)\right) \]

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. sinh-neg-rev N/A

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

      8. lower-sinh.f64 N/A

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

      9. lower-neg.f64 42.4

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

   6. Applied rewrites 42.4%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.086:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right) \cdot x\_m, x\_m, -0.005555555555555556\right) \cdot x\_m, x\_m, 0.16666666666666666\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sinh x\_m}{x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.086)
   (*
    (fma
     (*
      (fma
       (* (fma (* x_m x_m) -2.6455026455026456e-5 0.0003527336860670194) x_m)
       x_m
       -0.005555555555555556)
      x_m)
     x_m
     0.16666666666666666)
    (* x_m x_m))
   (log (/ (sinh x_m) x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.086) {
		tmp = fma((fma((fma((x_m * x_m), -2.6455026455026456e-5, 0.0003527336860670194) * x_m), x_m, -0.005555555555555556) * x_m), x_m, 0.16666666666666666) * (x_m * x_m);
	} else {
		tmp = log((sinh(x_m) / x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.086)
		tmp = Float64(fma(Float64(fma(Float64(fma(Float64(x_m * x_m), -2.6455026455026456e-5, 0.0003527336860670194) * x_m), x_m, -0.005555555555555556) * x_m), x_m, 0.16666666666666666) * Float64(x_m * x_m));
	else
		tmp = log(Float64(sinh(x_m) / x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.086], N[(N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.6455026455026456e-5 + 0.0003527336860670194), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + -0.005555555555555556), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + 0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Sinh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.085999999999999993

   1. Initial program 53.4%

      \[\log \left(\frac{\sinh x}{x}\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.6455026455026456 \cdot 10^{-5}, x \cdot x, 0.0003527336860670194\right) \cdot x\right) \cdot x - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot \left(x \cdot x\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{37800}, x \cdot x, \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{37800}, x \cdot x, \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{-1}{37800}, x \cdot x, \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}\right) \cdot {x}^{2} + \frac{1}{6}\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

      9. pow2 N/A

         \[\leadsto \left(\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}\right) \cdot x\right) \cdot x + \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\left(\frac{-1}{37800} \cdot \left(x \cdot x\right) + \frac{1}{2835}\right) \cdot x\right) \cdot x - \frac{1}{180}\right) \cdot x, x, \frac{1}{6}\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

   6. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right) \cdot x, x, -0.005555555555555556\right) \cdot x, x, 0.16666666666666666\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

   if 0.085999999999999993 < x

   1. Initial program 42.4%

      \[\log \left(\frac{\sinh x}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\_m\\ \frac{\left({\left(t\_0 \cdot x\_m\right)}^{2} - 0.027777777777777776\right) \cdot x\_m}{\mathsf{fma}\left(t\_0, x\_m, -0.16666666666666666\right)} \cdot x\_m \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (*
          (fma (* x_m x_m) 0.0003527336860670194 -0.005555555555555556)
          x_m)))
   (*
    (/
     (* (- (pow (* t_0 x_m) 2.0) 0.027777777777777776) x_m)
     (fma t_0 x_m -0.16666666666666666))
    x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma((x_m * x_m), 0.0003527336860670194, -0.005555555555555556) * x_m;
	return (((pow((t_0 * x_m), 2.0) - 0.027777777777777776) * x_m) / fma(t_0, x_m, -0.16666666666666666)) * x_m;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(fma(Float64(x_m * x_m), 0.0003527336860670194, -0.005555555555555556) * x_m)
	return Float64(Float64(Float64(Float64((Float64(t_0 * x_m) ^ 2.0) - 0.027777777777777776) * x_m) / fma(t_0, x_m, -0.16666666666666666)) * x_m)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(N[(N[(N[(N[Power[N[(t$95$0 * x$95$m), $MachinePrecision], 2.0], $MachinePrecision] - 0.027777777777777776), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(t$95$0 * x$95$m + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]

Derivation

1. Initial program 53.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot \color{blue}{x} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot \color{blue}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot {x}^{2} + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   9. lower--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   14. lower-*.f64 97.0

       \[\leadsto \left(\mathsf{fma}\left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x \]

4. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   3. lift--.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x \]

   9. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

   10. *-commutative N/A

       \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right) \cdot x \]

   12. +-commutative N/A

       \[\leadsto \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right)\right) \cdot x \]

   13. distribute-rgt-in N/A

       \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x + \frac{1}{6} \cdot x\right) \cdot x \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right), x, \frac{1}{6} \cdot x\right) \cdot x \]

6. Applied rewrites 97.0%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\right) \cdot x, x, 0.16666666666666666 \cdot x\right) \cdot x \]

8. Applied rewrites 97.1%

   \[\leadsto \frac{\left({\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\right) \cdot x\right)}^{2} - 0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x, x, -0.16666666666666666\right)} \cdot x \]
9. Add Preprocessing

Alternative 4: 97.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x\_m \cdot x\_m, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\_m\\ \frac{0.004629629629629629 \cdot x\_m}{\mathsf{fma}\left(t\_0 \cdot x\_m, \mathsf{fma}\left(t\_0, x\_m, -0.16666666666666666\right), 0.027777777777777776\right)} \cdot x\_m \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (*
          (fma (* x_m x_m) 0.0003527336860670194 -0.005555555555555556)
          x_m)))
   (*
    (/
     (* 0.004629629629629629 x_m)
     (fma (* t_0 x_m) (fma t_0 x_m -0.16666666666666666) 0.027777777777777776))
    x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma((x_m * x_m), 0.0003527336860670194, -0.005555555555555556) * x_m;
	return ((0.004629629629629629 * x_m) / fma((t_0 * x_m), fma(t_0, x_m, -0.16666666666666666), 0.027777777777777776)) * x_m;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(fma(Float64(x_m * x_m), 0.0003527336860670194, -0.005555555555555556) * x_m)
	return Float64(Float64(Float64(0.004629629629629629 * x_m) / fma(Float64(t_0 * x_m), fma(t_0, x_m, -0.16666666666666666), 0.027777777777777776)) * x_m)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(N[(N[(0.004629629629629629 * x$95$m), $MachinePrecision] / N[(N[(t$95$0 * x$95$m), $MachinePrecision] * N[(t$95$0 * x$95$m + -0.16666666666666666), $MachinePrecision] + 0.027777777777777776), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]

Derivation

1. Initial program 53.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot \color{blue}{x} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot \color{blue}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot {x}^{2} + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   9. lower--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   14. lower-*.f64 97.0

       \[\leadsto \left(\mathsf{fma}\left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x \]

4. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   3. lift--.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x \]

   9. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

   10. *-commutative N/A

       \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right) \cdot x \]

   12. +-commutative N/A

       \[\leadsto \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right)\right) \cdot x \]

   13. distribute-rgt-in N/A

       \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x + \frac{1}{6} \cdot x\right) \cdot x \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right), x, \frac{1}{6} \cdot x\right) \cdot x \]

6. Applied rewrites 97.0%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\right) \cdot x, x, 0.16666666666666666 \cdot x\right) \cdot x \]

8. Applied rewrites 97.1%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right)\right)}^{3}, {x}^{6}, 0.004629629629629629\right) \cdot x}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x, x, -0.16666666666666666\right), 0.027777777777777776\right)} \cdot x \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{1}{216} \cdot x}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{2835}, \frac{-1}{180}\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{2835}, \frac{-1}{180}\right) \cdot x, x, \frac{-1}{6}\right), \frac{1}{36}\right)} \cdot x \]
10. Step-by-step derivation

11. Applied rewrites 97.0%

    \[\leadsto \frac{0.004629629629629629 \cdot x}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x, x, -0.16666666666666666\right), 0.027777777777777776\right)} \cdot x \]
12. Add Preprocessing

Alternative 5: 97.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\_m\right) \cdot x\_m, x\_m, 0.16666666666666666 \cdot x\_m\right) \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (fma
   (*
    (* (fma (* x_m x_m) 0.0003527336860670194 -0.005555555555555556) x_m)
    x_m)
   x_m
   (* 0.16666666666666666 x_m))
  x_m))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(((fma((x_m * x_m), 0.0003527336860670194, -0.005555555555555556) * x_m) * x_m), x_m, (0.16666666666666666 * x_m)) * x_m;
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(fma(Float64(Float64(fma(Float64(x_m * x_m), 0.0003527336860670194, -0.005555555555555556) * x_m) * x_m), x_m, Float64(0.16666666666666666 * x_m)) * x_m)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m + N[(0.16666666666666666 * x$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot \color{blue}{x} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot \color{blue}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot {x}^{2} + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   9. lower--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   14. lower-*.f64 97.0

       \[\leadsto \left(\mathsf{fma}\left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x \]

4. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   3. lift--.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   6. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. +-commutative N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x \]

   8. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x \]

   9. pow2 N/A

      \[\leadsto \left(\left(\frac{1}{6} + \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

   10. *-commutative N/A

       \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot x \]

   11. *-commutative N/A

       \[\leadsto \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right) \cdot x \]

   12. +-commutative N/A

       \[\leadsto \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right)\right) \cdot x \]

   13. distribute-rgt-in N/A

       \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x + \frac{1}{6} \cdot x\right) \cdot x \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right), x, \frac{1}{6} \cdot x\right) \cdot x \]

6. Applied rewrites 97.0%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right) \cdot x\right) \cdot x, x, 0.16666666666666666 \cdot x\right) \cdot x \]
7. Add Preprocessing

Alternative 6: 97.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.0003527336860670194, -0.005555555555555556\right), x\_m \cdot x\_m, 0.16666666666666666\right) \cdot x\_m\right) \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (*
   (fma
    (fma (* x_m x_m) 0.0003527336860670194 -0.005555555555555556)
    (* x_m x_m)
    0.16666666666666666)
   x_m)
  x_m))

C

x_m = fabs(x);
double code(double x_m) {
	return (fma(fma((x_m * x_m), 0.0003527336860670194, -0.005555555555555556), (x_m * x_m), 0.16666666666666666) * x_m) * x_m;
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(fma(fma(Float64(x_m * x_m), 0.0003527336860670194, -0.005555555555555556), Float64(x_m * x_m), 0.16666666666666666) * x_m) * x_m)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot \color{blue}{x} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot \color{blue}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \cdot x\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \left(\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right) \cdot {x}^{2} + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   9. lower--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   11. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   13. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   14. lower-*.f64 97.0

       \[\leadsto \left(\mathsf{fma}\left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x \]

4. Applied rewrites 97.0%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0003527336860670194 \cdot \left(x \cdot x\right) - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
5. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot \left(x \cdot x\right) - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   4. pow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   5. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180} \cdot 1, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   6. fp-cancel-sub-sign-inv N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{1}{2835} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot 1, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{2835} + \left(\mathsf{neg}\left(\frac{1}{180}\right)\right) \cdot 1, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   8. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{2835} + \frac{-1}{180} \cdot 1, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   9. metadata-eval N/A

      \[\leadsto \left(\mathsf{fma}\left({x}^{2} \cdot \frac{1}{2835} + \frac{-1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{1}{2835}, \frac{-1}{180}\right), x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   11. pow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{2835}, \frac{-1}{180}\right), x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   12. lift-*.f64 97.0

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x \]

6. Applied rewrites 97.0%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003527336860670194, -0.005555555555555556\right), x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot \color{blue}{x} \]
7. Add Preprocessing

Alternative 7: 96.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(-0.005555555555555556, x\_m \cdot x\_m, 0.16666666666666666\right) \cdot \left(x\_m \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (fma -0.005555555555555556 (* x_m x_m) 0.16666666666666666) (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(-0.005555555555555556, (x_m * x_m), 0.16666666666666666) * (x_m * x_m);
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(fma(-0.005555555555555556, Float64(x_m * x_m), 0.16666666666666666) * Float64(x_m * x_m))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(-0.005555555555555556 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \cdot \color{blue}{{x}^{2}} \]

4. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.6455026455026456 \cdot 10^{-5}, x \cdot x, 0.0003527336860670194\right) \cdot x\right) \cdot x - 0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot \left(x \cdot x\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-1}{180}, x \cdot x, \frac{1}{6}\right) \cdot \left(x \cdot x\right) \]
6. Step-by-step derivation

7. Applied rewrites 96.6%

   \[\leadsto \mathsf{fma}\left(-0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot \left(x \cdot x\right) \]
8. Add Preprocessing

Alternative 8: 96.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(\mathsf{fma}\left(-0.005555555555555556, x\_m \cdot x\_m, 0.16666666666666666\right) \cdot x\_m\right) \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (* (fma -0.005555555555555556 (* x_m x_m) 0.16666666666666666) x_m) x_m))

C

x_m = fabs(x);
double code(double x_m) {
	return (fma(-0.005555555555555556, (x_m * x_m), 0.16666666666666666) * x_m) * x_m;
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(fma(-0.005555555555555556, Float64(x_m * x_m), 0.16666666666666666) * x_m) * x_m)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[(-0.005555555555555556 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot x\right) \cdot \color{blue}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right) \cdot x\right) \cdot x \]

   6. +-commutative N/A

      \[\leadsto \left(\left(\frac{-1}{180} \cdot {x}^{2} + \frac{1}{6}\right) \cdot x\right) \cdot x \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{180}, {x}^{2}, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   8. unpow2 N/A

      \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{180}, x \cdot x, \frac{1}{6}\right) \cdot x\right) \cdot x \]

   9. lower-*.f64 96.7

      \[\leadsto \left(\mathsf{fma}\left(-0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x \]

4. Applied rewrites 96.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.005555555555555556, x \cdot x, 0.16666666666666666\right) \cdot x\right) \cdot x} \]
5. Add Preprocessing

Alternative 9: 96.5% accurate, 19.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (* x_m x_m) 0.16666666666666666))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) * 0.16666666666666666;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = (x_m * x_m) * 0.16666666666666666d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) * 0.16666666666666666;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) * 0.16666666666666666

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) * 0.16666666666666666)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * x_m) * 0.16666666666666666;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 53.0%

   \[\log \left(\frac{\sinh x}{x}\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{6}} \]

   2. lower-*.f64 N/A

      \[\leadsto {x}^{2} \cdot \color{blue}{\frac{1}{6}} \]

   3. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \frac{1}{6} \]

   4. lower-*.f64 96.5

      \[\leadsto \left(x \cdot x\right) \cdot 0.16666666666666666 \]

4. Applied rewrites 96.5%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666} \]
5. Add Preprocessing

Developer Target 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.085:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6455026455026456 \cdot 10^{-5}, x \cdot x, 0.0003527336860670194\right), x \cdot x, -0.005555555555555556\right), x \cdot x, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sinh x}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.085)
   (*
    (* x x)
    (fma
     (fma
      (fma -2.6455026455026456e-5 (* x x) 0.0003527336860670194)
      (* x x)
      -0.005555555555555556)
     (* x x)
     0.16666666666666666))
   (log (/ (sinh x) x))))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.085) {
		tmp = (x * x) * fma(fma(fma(-2.6455026455026456e-5, (x * x), 0.0003527336860670194), (x * x), -0.005555555555555556), (x * x), 0.16666666666666666);
	} else {
		tmp = log((sinh(x) / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.085)
		tmp = Float64(Float64(x * x) * fma(fma(fma(-2.6455026455026456e-5, Float64(x * x), 0.0003527336860670194), Float64(x * x), -0.005555555555555556), Float64(x * x), 0.16666666666666666));
	else
		tmp = log(Float64(sinh(x) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.085], N[(N[(x * x), $MachinePrecision] * N[(N[(N[(-2.6455026455026456e-5 * N[(x * x), $MachinePrecision] + 0.0003527336860670194), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.005555555555555556), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]

Reproduce

herbie shell --seed 2025093 
(FPCore (x)
  :name "bug500, discussion (missed optimization)"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs x) 17/200) (let ((x2 (* x x))) (* x2 (fma (fma (fma -1/37800 x2 1/2835) x2 -1/180) x2 1/6))) (log (/ (sinh x) x))))

  (log (/ (sinh x) x)))