Hyperbolic sine

Percentage Accurate: 54.3% → 99.9%

Time: 3.2s

Alternatives: 8

Speedup: 2.5×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\frac{e^{x} - e^{-x}}{2} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

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 = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 54.3% accurate, 1.0× speedup

Math

\[\frac{e^{x} - e^{-x}}{2} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

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 = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.05:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \mathsf{fma}\left(0.008333333333333333, t\_0, 0.16666666666666666\right), t\_0, \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{2}{e^{\left|x\right|} - e^{-\left|x\right|}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 2.05)
     (fma
      (* (fabs x) (fma 0.008333333333333333 t_0 0.16666666666666666))
      t_0
      (fabs x))
     (/ 1.0 (/ 2.0 (- (exp (fabs x)) (exp (- (fabs x))))))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 2.05) {
		tmp = fma((fabs(x) * fma(0.008333333333333333, t_0, 0.16666666666666666)), t_0, fabs(x));
	} else {
		tmp = 1.0 / (2.0 / (exp(fabs(x)) - exp(-fabs(x))));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 2.05)
		tmp = fma(Float64(abs(x) * fma(0.008333333333333333, t_0, 0.16666666666666666)), t_0, abs(x));
	else
		tmp = Float64(1.0 / Float64(2.0 / Float64(exp(abs(x)) - exp(Float64(-abs(x))))));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 2.05], N[(N[(N[Abs[x], $MachinePrecision] * N[(0.008333333333333333 * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 / N[(N[Exp[N[Abs[x], $MachinePrecision]], $MachinePrecision] - N[Exp[(-N[Abs[x], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999998

   1. Initial program 54.3%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      7. lower-pow.f64 90.4%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{x \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}\right) + x \cdot 1 \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(x \cdot 1\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + \color{blue}{x} \cdot 1 \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, x \cdot 1\right) \]

   6. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, x\right) \]

   if 2.0499999999999998 < x

   1. Initial program 54.3%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 52.2%

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

   4. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

         \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(2 \cdot \color{blue}{x}\right) \cdot \frac{1}{2} \]

      5. count-2-rev N/A

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

      6. *-rgt-identity N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. *-rgt-identity N/A

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

      11. *-rgt-identity N/A

          \[\leadsto \left(x + x\right) \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{2}\right) \]

   6. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\left(x + x\right) \cdot 0.5} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{x + x}{2}} \]

      4. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{x + x}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{x + x}}} \]

      6. lower-unsound-/.f64 52.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{x + x}}} \]

   8. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{x + x}}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\frac{2}{\color{blue}{e^{x} - e^{\mathsf{neg}\left(x\right)}}}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-neg.f64 54.3%

          \[\leadsto \frac{1}{\frac{2}{e^{x} - e^{-x}}} \]

   11. Applied rewrites 54.3%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.05:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \mathsf{fma}\left(0.008333333333333333, t\_0, 0.16666666666666666\right), t\_0, \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{\left|x\right|} - e^{-\left|x\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 2.05)
     (fma
      (* (fabs x) (fma 0.008333333333333333 t_0 0.16666666666666666))
      t_0
      (fabs x))
     (* 0.5 (- (exp (fabs x)) (exp (- (fabs x)))))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 2.05) {
		tmp = fma((fabs(x) * fma(0.008333333333333333, t_0, 0.16666666666666666)), t_0, fabs(x));
	} else {
		tmp = 0.5 * (exp(fabs(x)) - exp(-fabs(x)));
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 2.05)
		tmp = fma(Float64(abs(x) * fma(0.008333333333333333, t_0, 0.16666666666666666)), t_0, abs(x));
	else
		tmp = Float64(0.5 * Float64(exp(abs(x)) - exp(Float64(-abs(x)))));
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 2.05], N[(N[(N[Abs[x], $MachinePrecision] * N[(0.008333333333333333 * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[N[Abs[x], $MachinePrecision]], $MachinePrecision] - N[Exp[(-N[Abs[x], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999998

   1. Initial program 54.3%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      7. lower-pow.f64 90.4%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{x \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}\right) + x \cdot 1 \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(x \cdot 1\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + \color{blue}{x} \cdot 1 \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, x \cdot 1\right) \]

   6. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, x\right) \]

   if 2.0499999999999998 < x

   1. Initial program 54.3%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 54.3%

         \[\leadsto 0.5 \cdot \left(e^{x} - e^{-x}\right) \]

   4. Applied rewrites 54.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{x} - e^{-x}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 110:\\ \;\;\;\;\mathsf{fma}\left(\left|x\right| \cdot \mathsf{fma}\left(0.008333333333333333, t\_0, 0.16666666666666666\right), t\_0, \left|x\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left|x\right|} - \left(1 - \left|x\right|\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1.0 x)
   (if (<= (fabs x) 110.0)
     (fma
      (* (fabs x) (fma 0.008333333333333333 t_0 0.16666666666666666))
      t_0
      (fabs x))
     (* (- (exp (fabs x)) (- 1.0 (fabs x))) 0.5)))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 110.0) {
		tmp = fma((fabs(x) * fma(0.008333333333333333, t_0, 0.16666666666666666)), t_0, fabs(x));
	} else {
		tmp = (exp(fabs(x)) - (1.0 - fabs(x))) * 0.5;
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 110.0)
		tmp = fma(Float64(abs(x) * fma(0.008333333333333333, t_0, 0.16666666666666666)), t_0, abs(x));
	else
		tmp = Float64(Float64(exp(abs(x)) - Float64(1.0 - abs(x))) * 0.5);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 110.0], N[(N[(N[Abs[x], $MachinePrecision] * N[(0.008333333333333333 * t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[Abs[x], $MachinePrecision]], $MachinePrecision] - N[(1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 110

   1. Initial program 54.3%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot \color{blue}{{x}^{2}}\right)\right) \]

      7. lower-pow.f64 90.4%

         \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right) \]

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

      3. +-commutative N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right) \]

      4. distribute-lft-in N/A

         \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{x \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

      7. *-commutative N/A

         \[\leadsto \left(x \cdot 1\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}\right) + x \cdot 1 \]

      8. associate-*r* N/A

         \[\leadsto \left(\left(x \cdot 1\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + \color{blue}{x} \cdot 1 \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, x \cdot 1\right) \]

   6. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, x\right) \]

   if 110 < x

   1. Initial program 54.3%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 30.4%

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

   4. Applied rewrites 30.4%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-flip-reverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. *-rgt-identity N/A

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

      11. *-rgt-identity N/A

          \[\leadsto \left(e^{x} - \left(1 - x\right)\right) \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{2}\right) \]

   6. Applied rewrites 30.4%

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

Alternative 4: 90.4% accurate, 1.5× speedup

Math

\[\mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), x \cdot x, x\right) \]

FPCore

(FPCore (x)
  :precision binary64
  (fma
 (* x (fma 0.008333333333333333 (* x x) 0.16666666666666666))
 (* x x)
 x))

C

double code(double x) {
	return fma((x * fma(0.008333333333333333, (x * x), 0.16666666666666666)), (x * x), x);
}

Julia

function code(x)
	return fma(Float64(x * fma(0.008333333333333333, Float64(x * x), 0.16666666666666666)), Float64(x * x), x)
end

Wolfram

code[x_] := N[(N[(x * N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {x}^{2}\right)\right) \]

   5. lower-+.f64 N/A

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{120} \cdot {x}^{2}}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot \color{blue}{{x}^{2}}\right)\right) \]

   7. lower-pow.f64 90.4%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right) \]

4. Applied rewrites 90.4%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto x \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)}\right) \]

   3. +-commutative N/A

      \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + \color{blue}{1}\right) \]

   4. distribute-lft-in N/A

      \[\leadsto x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + \color{blue}{x \cdot 1} \]

   5. *-rgt-identity N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) + x \cdot 1 \]

   7. *-commutative N/A

      \[\leadsto \left(x \cdot 1\right) \cdot \left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}\right) + x \cdot 1 \]

   8. associate-*r* N/A

      \[\leadsto \left(\left(x \cdot 1\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + \color{blue}{x} \cdot 1 \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot 1\right) \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, x \cdot 1\right) \]

6. Applied rewrites 90.4%

   \[\leadsto \mathsf{fma}\left(x \cdot \mathsf{fma}\left(0.008333333333333333, x \cdot x, 0.16666666666666666\right), \color{blue}{x \cdot x}, x\right) \]
7. Add Preprocessing

Alternative 5: 83.8% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (fma (* x x) (* 0.16666666666666666 x) x))

C

double code(double x) {
	return fma((x * x), (0.16666666666666666 * x), x);
}

Julia

function code(x)
	return fma(Float64(x * x), Float64(0.16666666666666666 * x), x)
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 83.8%

      \[\leadsto x \cdot \left(1 + 0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) \]

4. Applied rewrites 83.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot {x}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. +-commutative N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. pow2 N/A

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

   14. lower-*.f64 N/A

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

   15. *-rgt-identity N/A

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

   16. *-rgt-identity N/A

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

   17. lower-*.f64 N/A

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

   18. *-rgt-identity N/A

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

   19. *-rgt-identity 83.8%

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

6. Applied rewrites 83.8%

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

Alternative 6: 83.8% accurate, 2.5× speedup

Math

\[x \cdot \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \]

FPCore

(FPCore (x)
  :precision binary64
  (* x (fma (* 0.16666666666666666 x) x 1.0)))

C

double code(double x) {
	return x * fma((0.16666666666666666 * x), x, 1.0);
}

Julia

function code(x)
	return Float64(x * fma(Float64(0.16666666666666666 * x), x, 1.0))
end

Wolfram

code[x_] := N[(x * N[(N[(0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 83.8%

      \[\leadsto x \cdot \left(1 + 0.16666666666666666 \cdot {x}^{\color{blue}{2}}\right) \]

4. Applied rewrites 83.8%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot {x}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. *-rgt-identity N/A

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

   6. pow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. *-rgt-identity N/A

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

   11. *-rgt-identity 83.8%

       \[\leadsto x \cdot \mathsf{fma}\left(0.16666666666666666 \cdot x, x, 1\right) \]

6. Applied rewrites 83.8%

   \[\leadsto x \cdot \mathsf{fma}\left(0.16666666666666666 \cdot x, \color{blue}{x}, 1\right) \]
7. Add Preprocessing

Alternative 7: 52.2% accurate, 2.9× speedup

Math

\[\frac{\left(x + x\right) \cdot 2}{4} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ (* (+ x x) 2.0) 4.0))

C

double code(double x) {
	return ((x + x) * 2.0) / 4.0;
}

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 = ((x + x) * 2.0d0) / 4.0d0
end function

Java

public static double code(double x) {
	return ((x + x) * 2.0) / 4.0;
}

Python

def code(x):
	return ((x + x) * 2.0) / 4.0

Julia

function code(x)
	return Float64(Float64(Float64(x + x) * 2.0) / 4.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x + x) * 2.0) / 4.0;
end

Wolfram

code[x_] := N[(N[(N[(x + x), $MachinePrecision] * 2.0), $MachinePrecision] / 4.0), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0

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

   1. lower-*.f64 52.2%

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

4. Applied rewrites 52.2%

   \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \color{blue}{x}\right) \cdot \frac{1}{2} \]

   5. count-2-rev N/A

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

   6. *-rgt-identity N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f64 N/A

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

   9. *-rgt-identity N/A

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

   10. *-rgt-identity N/A

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

   11. *-rgt-identity N/A

       \[\leadsto \left(x + x\right) \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{2}\right) \]

6. Applied rewrites 52.2%

   \[\leadsto \color{blue}{\left(x + x\right) \cdot 0.5} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{x + x}{2}} \]

   4. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{x + x}}} \]

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

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{x + x}}} \]

   6. lower-unsound-/.f64 52.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{2}{x + x}}} \]

8. Applied rewrites 52.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{2}{x + x}}} \]
9. Step-by-step derivation

   1. *-lft-identity N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-flip-rev N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{x + x}{2} \cdot \color{blue}{\frac{2}{2}} \]

   7. frac-times N/A

      \[\leadsto \color{blue}{\frac{\left(x + x\right) \cdot 2}{2 \cdot 2}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(x + x\right) \cdot 2}{2 \cdot 2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(x + x\right) \cdot 2}}{2 \cdot 2} \]

   10. metadata-eval 52.2%

       \[\leadsto \frac{\left(x + x\right) \cdot 2}{\color{blue}{4}} \]

10. Applied rewrites 52.2%

    \[\leadsto \color{blue}{\frac{\left(x + x\right) \cdot 2}{4}} \]
11. Add Preprocessing

Alternative 8: 52.2% accurate, 4.6× speedup

Math

\[\left(x + x\right) \cdot 0.5 \]

FPCore

(FPCore (x)
  :precision binary64
  (* (+ x x) 0.5))

C

double code(double x) {
	return (x + x) * 0.5;
}

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 = (x + x) * 0.5d0
end function

Java

public static double code(double x) {
	return (x + x) * 0.5;
}

Python

def code(x):
	return (x + x) * 0.5

Julia

function code(x)
	return Float64(Float64(x + x) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x + x) * 0.5;
end

Wolfram

code[x_] := N[(N[(x + x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0

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

   1. lower-*.f64 52.2%

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

4. Applied rewrites 52.2%

   \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{2}} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot \color{blue}{x}\right) \cdot \frac{1}{2} \]

   5. count-2-rev N/A

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

   6. *-rgt-identity N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f64 N/A

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

   9. *-rgt-identity N/A

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

   10. *-rgt-identity N/A

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

   11. *-rgt-identity N/A

       \[\leadsto \left(x + x\right) \cdot \mathsf{Rewrite=>}\left(metadata-eval, \frac{1}{2}\right) \]

6. Applied rewrites 52.2%

   \[\leadsto \color{blue}{\left(x + x\right) \cdot 0.5} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025245 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))