Hyperbolic sine

Percentage Accurate: 55.3% → 99.8%

Time: 1.9min

Alternatives: 9

Speedup: 11.4×

• 49.2% 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))

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), $MachinePrecision]

Initial Program: 55.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))

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), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq \frac{19}{2}:\\ \;\;\;\;\left(\left|x\right| \cdot \left(\frac{1}{120} \cdot t\_0 - \frac{-1}{6}\right)\right) \cdot t\_0 + \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(e^{\left|x\right|} - e^{-\left|x\right|}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1 x)
   (if (<= (fabs x) 19/2)
     (+ (* (* (fabs x) (- (* 1/120 t_0) -1/6)) t_0) (fabs x))
     (* 1/2 (- (exp (fabs x)) (exp (- (fabs x)))))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 9.5) {
		tmp = ((fabs(x) * ((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;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 9.5) {
		tmp = ((Math.abs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + Math.abs(x);
	} else {
		tmp = 0.5 * (Math.exp(Math.abs(x)) - Math.exp(-Math.abs(x)));
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 9.5:
		tmp = ((math.fabs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + math.fabs(x)
	else:
		tmp = 0.5 * (math.exp(math.fabs(x)) - math.exp(-math.fabs(x)))
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 9.5)
		tmp = Float64(Float64(Float64(abs(x) * Float64(Float64(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

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs(x);
	tmp = 0.0;
	if (abs(x) <= 9.5)
		tmp = ((abs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + abs(x);
	else
		tmp = 0.5 * (exp(abs(x)) - exp(-abs(x)));
	end
	tmp_2 = (sign(x) * abs(1.0)) * 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], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 19/2], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(1/120 * t$95$0), $MachinePrecision] - -1/6), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(1/2 * 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 < 9.5

   1. Initial program 55.3%

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

   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 90.7%

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   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-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

   6. Applied rewrites 90.7%

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

   if 9.5 < x

   1. Initial program 55.3%

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

   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 90.7%

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   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-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

   6. Applied rewrites 90.7%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{x} - e^{\mathsf{neg}\left(x\right)}\right)} \]
   8. 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 55.3%

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

   9. Applied rewrites 55.3%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq \frac{19}{2}:\\ \;\;\;\;\left(\left|x\right| \cdot \left(\frac{1}{120} \cdot t\_0 - \frac{-1}{6}\right)\right) \cdot t\_0 + \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left|x\right|} - \left(1 - \left|x\right|\right)\right) \cdot \frac{1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))))
  (*
   (copysign 1 x)
   (if (<= (fabs x) 19/2)
     (+ (* (* (fabs x) (- (* 1/120 t_0) -1/6)) t_0) (fabs x))
     (* (- (exp (fabs x)) (- 1 (fabs x))) 1/2)))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double tmp;
	if (fabs(x) <= 9.5) {
		tmp = ((fabs(x) * ((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;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 9.5) {
		tmp = ((Math.abs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + Math.abs(x);
	} else {
		tmp = (Math.exp(Math.abs(x)) - (1.0 - Math.abs(x))) * 0.5;
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 9.5:
		tmp = ((math.fabs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + math.fabs(x)
	else:
		tmp = (math.exp(math.fabs(x)) - (1.0 - math.fabs(x))) * 0.5
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	tmp = 0.0
	if (abs(x) <= 9.5)
		tmp = Float64(Float64(Float64(abs(x) * Float64(Float64(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

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs(x);
	tmp = 0.0;
	if (abs(x) <= 9.5)
		tmp = ((abs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + abs(x);
	else
		tmp = (exp(abs(x)) - (1.0 - abs(x))) * 0.5;
	end
	tmp_2 = (sign(x) * abs(1.0)) * 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], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 19/2], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(1/120 * t$95$0), $MachinePrecision] - -1/6), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[Abs[x], $MachinePrecision]], $MachinePrecision] - N[(1 - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5

   1. Initial program 55.3%

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

   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 90.7%

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   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-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

   6. Applied rewrites 90.7%

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

   if 9.5 < x

   1. Initial program 55.3%

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

   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 30.9%

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

   4. Applied rewrites 30.9%

      \[\leadsto \frac{e^{x} - \color{blue}{\left(1 + -1 \cdot x\right)}}{2} \]
   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 30.9%

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

Alternative 3: 94.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 25999999999999999036901235468997230592:\\ \;\;\;\;\left(\left|x\right| \cdot \left(\frac{1}{120} \cdot t\_0 - \frac{-1}{6}\right)\right) \cdot t\_0 + \left|x\right|\\ \mathbf{elif}\;\left|x\right| \leq 10000000000000000019156750857346687362159551272651920111528035145993793242039887559612361451081803235328:\\ \;\;\;\;\frac{\left|x\right| \cdot \left(\left(\frac{1}{36} \cdot \sqrt{t\_1 \cdot t\_1}\right) \cdot \left|x\right| - \left|x\right|\right)}{\left(\frac{1}{6} \cdot t\_0\right) \cdot \left|x\right| - \left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left(1 + \left(\frac{1}{6} \cdot \left|x\right|\right) \cdot \left|x\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x)))
       (t_1 (* (* t_0 (fabs x)) (fabs x))))
  (*
   (copysign 1 x)
   (if (<= (fabs x) 25999999999999999036901235468997230592)
     (+ (* (* (fabs x) (- (* 1/120 t_0) -1/6)) t_0) (fabs x))
     (if (<=
          (fabs x)
          10000000000000000019156750857346687362159551272651920111528035145993793242039887559612361451081803235328)
       (/
        (*
         (fabs x)
         (- (* (* 1/36 (sqrt (* t_1 t_1))) (fabs x)) (fabs x)))
        (- (* (* 1/6 t_0) (fabs x)) (fabs x)))
       (* (fabs x) (+ 1 (* (* 1/6 (fabs x)) (fabs x)))))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(x) <= 2.6e+37) {
		tmp = ((fabs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + fabs(x);
	} else if (fabs(x) <= 1e+103) {
		tmp = (fabs(x) * (((0.027777777777777776 * sqrt((t_1 * t_1))) * fabs(x)) - fabs(x))) / (((0.16666666666666666 * t_0) * fabs(x)) - fabs(x));
	} else {
		tmp = fabs(x) * (1.0 + ((0.16666666666666666 * fabs(x)) * fabs(x)));
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 2.6e+37) {
		tmp = ((Math.abs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + Math.abs(x);
	} else if (Math.abs(x) <= 1e+103) {
		tmp = (Math.abs(x) * (((0.027777777777777776 * Math.sqrt((t_1 * t_1))) * Math.abs(x)) - Math.abs(x))) / (((0.16666666666666666 * t_0) * Math.abs(x)) - Math.abs(x));
	} else {
		tmp = Math.abs(x) * (1.0 + ((0.16666666666666666 * Math.abs(x)) * Math.abs(x)));
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 2.6e+37:
		tmp = ((math.fabs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + math.fabs(x)
	elif math.fabs(x) <= 1e+103:
		tmp = (math.fabs(x) * (((0.027777777777777776 * math.sqrt((t_1 * t_1))) * math.fabs(x)) - math.fabs(x))) / (((0.16666666666666666 * t_0) * math.fabs(x)) - math.fabs(x))
	else:
		tmp = math.fabs(x) * (1.0 + ((0.16666666666666666 * math.fabs(x)) * math.fabs(x)))
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(x) <= 2.6e+37)
		tmp = Float64(Float64(Float64(abs(x) * Float64(Float64(0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + abs(x));
	elseif (abs(x) <= 1e+103)
		tmp = Float64(Float64(abs(x) * Float64(Float64(Float64(0.027777777777777776 * sqrt(Float64(t_1 * t_1))) * abs(x)) - abs(x))) / Float64(Float64(Float64(0.16666666666666666 * t_0) * abs(x)) - abs(x)));
	else
		tmp = Float64(abs(x) * Float64(1.0 + Float64(Float64(0.16666666666666666 * abs(x)) * abs(x))));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(x) <= 2.6e+37)
		tmp = ((abs(x) * ((0.008333333333333333 * t_0) - -0.16666666666666666)) * t_0) + abs(x);
	elseif (abs(x) <= 1e+103)
		tmp = (abs(x) * (((0.027777777777777776 * sqrt((t_1 * t_1))) * abs(x)) - abs(x))) / (((0.16666666666666666 * t_0) * abs(x)) - abs(x));
	else
		tmp = abs(x) * (1.0 + ((0.16666666666666666 * abs(x)) * abs(x)));
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 25999999999999999036901235468997230592], N[(N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(1/120 * t$95$0), $MachinePrecision] - -1/6), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 10000000000000000019156750857346687362159551272651920111528035145993793242039887559612361451081803235328], N[(N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(1/36 * N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1/6 * t$95$0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(1 + N[(N[(1/6 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5999999999999999e37

   1. Initial program 55.3%

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

   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 90.7%

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

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   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-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. lower-+.f64 N/A

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

   6. Applied rewrites 90.7%

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

   if 2.5999999999999999e37 < x < 1e103

   1. Initial program 55.3%

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

   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 84.1%

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

   4. Applied rewrites 84.1%

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
   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. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-rgt-identity N/A

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

      6. *-lft-identity N/A

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

      7. flip-+ N/A

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

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

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

   6. Applied rewrites 35.9%

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

   8. Applied rewrites 35.9%

      \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot x - x\right)}{\color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x}} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \left(\sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} \cdot \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      2. sqrt-unprod N/A

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      4. lower-*.f64 37.8%

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      7. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      9. pow3 N/A

         \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      10. lower-pow.f32 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      11. lower-unsound-pow.f32 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      13. lower-unsound-pow.f32 41.5%

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left( {x}^{3} \right)_{\text{binary32}} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      14. lower-pow.f32 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left({x}^{3} \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      15. pow3 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      17. lift-*.f64 37.8%

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      20. associate-*r* N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      22. pow3 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left({x}^{3} \cdot x\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      23. lower-pow.f32 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left({x}^{3} \cdot x\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      24. lower-unsound-pow.f32 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left({x}^{3} \cdot x\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

      25. lower-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left({x}^{3} \cdot x\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

   10. Applied rewrites 37.8%

       \[\leadsto \frac{x \cdot \left(\left(\frac{1}{36} \cdot \sqrt{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)}\right) \cdot x - x\right)}{\left(\frac{1}{6} \cdot \left(x \cdot x\right)\right) \cdot x - x} \]

   if 1e103 < x

   1. Initial program 55.3%

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

   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 84.1%

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

   4. Applied rewrites 84.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. *-rgt-identity 84.1%

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

   6. Applied rewrites 84.1%

      \[\leadsto x \cdot \left(1 + \left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{x}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := \frac{1}{120} \cdot t\_0 - \frac{-1}{6}\\ t_2 := \left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 39999999999999997975485411882960754678545800440536402923356160:\\ \;\;\;\;\left|x\right| \cdot \frac{t\_2 \cdot t\_2 - 1 \cdot 1}{t\_2 - 1}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left(1 + t\_1 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x)))
       (t_1 (- (* 1/120 t_0) -1/6))
       (t_2 (* (* t_1 (fabs x)) (fabs x))))
  (*
   (copysign 1 x)
   (if (<=
        (fabs x)
        39999999999999997975485411882960754678545800440536402923356160)
     (* (fabs x) (/ (- (* t_2 t_2) (* 1 1)) (- t_2 1)))
     (* (fabs x) (+ 1 (* t_1 t_0)))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = (0.008333333333333333 * t_0) - -0.16666666666666666;
	double t_2 = (t_1 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs(x) <= 4e+61) {
		tmp = fabs(x) * (((t_2 * t_2) - (1.0 * 1.0)) / (t_2 - 1.0));
	} else {
		tmp = fabs(x) * (1.0 + (t_1 * t_0));
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double t_1 = (0.008333333333333333 * t_0) - -0.16666666666666666;
	double t_2 = (t_1 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs(x) <= 4e+61) {
		tmp = Math.abs(x) * (((t_2 * t_2) - (1.0 * 1.0)) / (t_2 - 1.0));
	} else {
		tmp = Math.abs(x) * (1.0 + (t_1 * t_0));
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = (0.008333333333333333 * t_0) - -0.16666666666666666
	t_2 = (t_1 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs(x) <= 4e+61:
		tmp = math.fabs(x) * (((t_2 * t_2) - (1.0 * 1.0)) / (t_2 - 1.0))
	else:
		tmp = math.fabs(x) * (1.0 + (t_1 * t_0))
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = Float64(Float64(0.008333333333333333 * t_0) - -0.16666666666666666)
	t_2 = Float64(Float64(t_1 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(x) <= 4e+61)
		tmp = Float64(abs(x) * Float64(Float64(Float64(t_2 * t_2) - Float64(1.0 * 1.0)) / Float64(t_2 - 1.0)));
	else
		tmp = Float64(abs(x) * Float64(1.0 + Float64(t_1 * t_0)));
	end
	return Float64(copysign(1.0, x) * tmp)
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs(x);
	t_1 = (0.008333333333333333 * t_0) - -0.16666666666666666;
	t_2 = (t_1 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs(x) <= 4e+61)
		tmp = abs(x) * (((t_2 * t_2) - (1.0 * 1.0)) / (t_2 - 1.0));
	else
		tmp = abs(x) * (1.0 + (t_1 * t_0));
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1/120 * t$95$0), $MachinePrecision] - -1/6), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 39999999999999997975485411882960754678545800440536402923356160], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(1 * 1), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(1 + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999998e61

   1. Initial program 55.3%

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

   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 90.7%

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

   4. Applied rewrites 90.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   6. Applied rewrites 55.2%

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

   if 3.9999999999999998e61 < x

   1. Initial program 55.3%

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

   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 90.7%

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

   4. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.7%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-rgt-identity N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval 90.7%

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

      15. lift-pow.f64 N/A

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

      16. *-rgt-identity N/A

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

      17. pow2 N/A

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

      18. lower-*.f64 N/A

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

      19. *-rgt-identity N/A

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

      20. *-rgt-identity 90.7%

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

   6. Applied rewrites 90.7%

      \[\leadsto x \cdot \left(1 + \left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{-1}{6}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.7% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (+ (* (* x (- (* 1/120 (* x x)) -1/6)) (* x x)) x))

C

double code(double x) {
	return ((x * ((0.008333333333333333 * (x * x)) - -0.16666666666666666)) * (x * 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 = ((x * ((0.008333333333333333d0 * (x * x)) - (-0.16666666666666666d0))) * (x * x)) + x
end function

Java

public static double code(double x) {
	return ((x * ((0.008333333333333333 * (x * x)) - -0.16666666666666666)) * (x * x)) + x;
}

Python

def code(x):
	return ((x * ((0.008333333333333333 * (x * x)) - -0.16666666666666666)) * (x * x)) + x

Julia

function code(x)
	return Float64(Float64(Float64(x * Float64(Float64(0.008333333333333333 * Float64(x * x)) - -0.16666666666666666)) * Float64(x * x)) + x)
end

MATLAB

function tmp = code(x)
	tmp = ((x * ((0.008333333333333333 * (x * x)) - -0.16666666666666666)) * (x * x)) + x;
end

Wolfram

code[x_] := N[(N[(N[(x * N[(N[(1/120 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1/6), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

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 90.7%

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

4. Applied rewrites 90.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
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-rgt-in N/A

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

   5. *-rgt-identity N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 N/A

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

6. Applied rewrites 90.7%

   \[\leadsto \left(x \cdot \left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{-1}{6}\right)\right) \cdot \left(x \cdot x\right) + \color{blue}{x} \]
7. Add Preprocessing

Alternative 6: 90.7% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* x (+ 1 (* (- (* 1/120 (* x x)) -1/6) (* x x)))))

C

double code(double x) {
	return x * (1.0 + (((0.008333333333333333 * (x * x)) - -0.16666666666666666) * (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 = x * (1.0d0 + (((0.008333333333333333d0 * (x * x)) - (-0.16666666666666666d0)) * (x * x)))
end function

Java

public static double code(double x) {
	return x * (1.0 + (((0.008333333333333333 * (x * x)) - -0.16666666666666666) * (x * x)));
}

Python

def code(x):
	return x * (1.0 + (((0.008333333333333333 * (x * x)) - -0.16666666666666666) * (x * x)))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(Float64(0.008333333333333333 * Float64(x * x)) - -0.16666666666666666) * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (((0.008333333333333333 * (x * x)) - -0.16666666666666666) * (x * x)));
end

Wolfram

code[x_] := N[(x * N[(1 + N[(N[(N[(1/120 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1/6), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

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 90.7%

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

4. Applied rewrites 90.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 90.7%

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. add-flip N/A

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

   7. lower--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. *-rgt-identity N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. *-rgt-identity N/A

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

   14. metadata-eval 90.7%

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

   15. lift-pow.f64 N/A

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

   16. *-rgt-identity N/A

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

   17. pow2 N/A

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

   18. lower-*.f64 N/A

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

   19. *-rgt-identity N/A

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

   20. *-rgt-identity 90.7%

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

6. Applied rewrites 90.7%

   \[\leadsto x \cdot \left(1 + \left(\frac{1}{120} \cdot \left(x \cdot x\right) - \frac{-1}{6}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]
7. Add Preprocessing

Alternative 7: 84.1% accurate, 11.4× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (+ (* (* x 1/6) (* x x)) x))

C

double code(double x) {
	return ((x * 0.16666666666666666) * (x * 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 = ((x * 0.16666666666666666d0) * (x * x)) + x
end function

Java

public static double code(double x) {
	return ((x * 0.16666666666666666) * (x * x)) + x;
}

Python

def code(x):
	return ((x * 0.16666666666666666) * (x * x)) + x

Julia

function code(x)
	return Float64(Float64(Float64(x * 0.16666666666666666) * Float64(x * x)) + x)
end

MATLAB

function tmp = code(x)
	tmp = ((x * 0.16666666666666666) * (x * x)) + x;
end

Wolfram

code[x_] := N[(N[(N[(x * 1/6), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

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 90.7%

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

4. Applied rewrites 90.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
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-rgt-in N/A

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

   5. *-rgt-identity N/A

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

   6. *-lft-identity N/A

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

   7. lower-+.f64 N/A

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

6. Applied rewrites 90.7%

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

7. Taylor expanded in x around 0

   \[\leadsto \left(x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right) + x \]
8. Step-by-step derivation

9. Applied rewrites 84.1%

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

Alternative 8: 84.1% accurate, 11.4× speedup

Math

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

FPCore

(FPCore (x)
  :precision binary64
  (* x (+ 1 (* (* 1/6 x) x))))

C

double code(double x) {
	return x * (1.0 + ((0.16666666666666666 * 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 = x * (1.0d0 + ((0.16666666666666666d0 * x) * x))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((0.16666666666666666 * x) * x));
}

Python

def code(x):
	return x * (1.0 + ((0.16666666666666666 * x) * x))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(0.16666666666666666 * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((0.16666666666666666 * x) * x));
end

Wolfram

code[x_] := N[(x * N[(1 + N[(N[(1/6 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

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 84.1%

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

4. Applied rewrites 84.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. pow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. *-rgt-identity N/A

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

   9. *-rgt-identity 84.1%

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

6. Applied rewrites 84.1%

   \[\leadsto x \cdot \left(1 + \left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{x}\right) \]
7. Add Preprocessing

Alternative 9: 51.0% accurate, 36.2× speedup

Math

\[x \cdot 1 \]

FPCore

(FPCore (x)
  :precision binary64
  (* x 1))

C

double code(double x) {
	return x * 1.0;
}

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 * 1.0d0
end function

Java

public static double code(double x) {
	return x * 1.0;
}

Python

def code(x):
	return x * 1.0

Julia

function code(x)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x)
	tmp = x * 1.0;
end

Wolfram

code[x_] := N[(x * 1), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

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 90.7%

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

4. Applied rewrites 90.7%

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

5. Taylor expanded in x around 0

   \[\leadsto x \cdot 1 \]
6. Step-by-step derivation

7. Applied rewrites 51.0%

   \[\leadsto x \cdot 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))