bug500, discussion (missed optimization)

Percentage Accurate: 53.0% → 97.5%

Time: 24.3s

Alternatives: 6

Speedup: 2.0×

• could not determine a ground truth

  1. x = 108624639857119790.0

Specification

Math

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

FPCore

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

C

double code(double x) {
	return log((sinh(x) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((sinh(x) / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return log((sinh(x) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((sinh(x) / x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh x\_m}{x\_m} \leq 1.005:\\ \;\;\;\;{x\_m}^{2} \cdot \left(0.16666666666666666 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 0.0003527336860670194 - 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \sinh x\_m - \log x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (/ (sinh x_m) x_m) 1.005)
   (*
    (pow x_m 2.0)
    (+
     0.16666666666666666
     (*
      (pow x_m 2.0)
      (- (* (pow x_m 2.0) 0.0003527336860670194) 0.005555555555555556))))
   (- (log (sinh x_m)) (log x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((sinh(x_m) / x_m) <= 1.005) {
		tmp = pow(x_m, 2.0) * (0.16666666666666666 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.0003527336860670194) - 0.005555555555555556)));
	} else {
		tmp = log(sinh(x_m)) - log(x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if ((sinh(x_m) / x_m) <= 1.005d0) then
        tmp = (x_m ** 2.0d0) * (0.16666666666666666d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.0003527336860670194d0) - 0.005555555555555556d0)))
    else
        tmp = log(sinh(x_m)) - log(x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if ((Math.sinh(x_m) / x_m) <= 1.005) {
		tmp = Math.pow(x_m, 2.0) * (0.16666666666666666 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.0003527336860670194) - 0.005555555555555556)));
	} else {
		tmp = Math.log(Math.sinh(x_m)) - Math.log(x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if (math.sinh(x_m) / x_m) <= 1.005:
		tmp = math.pow(x_m, 2.0) * (0.16666666666666666 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 0.0003527336860670194) - 0.005555555555555556)))
	else:
		tmp = math.log(math.sinh(x_m)) - math.log(x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(sinh(x_m) / x_m) <= 1.005)
		tmp = Float64((x_m ^ 2.0) * Float64(0.16666666666666666 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.0003527336860670194) - 0.005555555555555556))));
	else
		tmp = Float64(log(sinh(x_m)) - log(x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if ((sinh(x_m) / x_m) <= 1.005)
		tmp = (x_m ^ 2.0) * (0.16666666666666666 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 0.0003527336860670194) - 0.005555555555555556)));
	else
		tmp = log(sinh(x_m)) - log(x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(N[Sinh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision], 1.005], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.0003527336860670194), $MachinePrecision] - 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[Sinh[x$95$m], $MachinePrecision]], $MachinePrecision] - N[Log[x$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 x) x) < 1.0049999999999999

   1. Initial program 51.0%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left(0.0003527336860670194 \cdot {x}^{2} - 0.005555555555555556\right)\right)} \]

   if 1.0049999999999999 < (/.f64 (sinh.f64 x) x)

   1. Initial program 76.8%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. log-div 44.1%

         \[\leadsto \color{blue}{\log \sinh x - \log x} \]

      2. sub-neg 44.1%

         \[\leadsto \color{blue}{\log \sinh x + \left(-\log x\right)} \]

   4. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{\log \sinh x + \left(-\log x\right)} \]
   5. Step-by-step derivation

      1. sub-neg 44.1%

         \[\leadsto \color{blue}{\log \sinh x - \log x} \]

   6. Simplified 44.1%

      \[\leadsto \color{blue}{\log \sinh x - \log x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.005:\\ \;\;\;\;{x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0003527336860670194 - 0.005555555555555556\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \sinh x - \log x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sinh x\_m}{x\_m}\\ \mathbf{if}\;t\_0 \leq 1.00001:\\ \;\;\;\;{x\_m}^{2} \cdot \left(0.16666666666666666 + {x\_m}^{2} \cdot -0.005555555555555556\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t\_0 + -1\right) + 1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ (sinh x_m) x_m)))
   (if (<= t_0 1.00001)
     (*
      (pow x_m 2.0)
      (+ 0.16666666666666666 (* (pow x_m 2.0) -0.005555555555555556)))
     (+ (+ (log t_0) -1.0) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sinh(x_m) / x_m;
	double tmp;
	if (t_0 <= 1.00001) {
		tmp = pow(x_m, 2.0) * (0.16666666666666666 + (pow(x_m, 2.0) * -0.005555555555555556));
	} else {
		tmp = (log(t_0) + -1.0) + 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(x_m) / x_m
    if (t_0 <= 1.00001d0) then
        tmp = (x_m ** 2.0d0) * (0.16666666666666666d0 + ((x_m ** 2.0d0) * (-0.005555555555555556d0)))
    else
        tmp = (log(t_0) + (-1.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sinh(x_m) / x_m;
	double tmp;
	if (t_0 <= 1.00001) {
		tmp = Math.pow(x_m, 2.0) * (0.16666666666666666 + (Math.pow(x_m, 2.0) * -0.005555555555555556));
	} else {
		tmp = (Math.log(t_0) + -1.0) + 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sinh(x_m) / x_m
	tmp = 0
	if t_0 <= 1.00001:
		tmp = math.pow(x_m, 2.0) * (0.16666666666666666 + (math.pow(x_m, 2.0) * -0.005555555555555556))
	else:
		tmp = (math.log(t_0) + -1.0) + 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(sinh(x_m) / x_m)
	tmp = 0.0
	if (t_0 <= 1.00001)
		tmp = Float64((x_m ^ 2.0) * Float64(0.16666666666666666 + Float64((x_m ^ 2.0) * -0.005555555555555556)));
	else
		tmp = Float64(Float64(log(t_0) + -1.0) + 1.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = sinh(x_m) / x_m;
	tmp = 0.0;
	if (t_0 <= 1.00001)
		tmp = (x_m ^ 2.0) * (0.16666666666666666 + ((x_m ^ 2.0) * -0.005555555555555556));
	else
		tmp = (log(t_0) + -1.0) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Sinh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 1.00001], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 x) x) < 1.0000100000000001

   1. Initial program 50.9%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.16666666666666666 + -0.005555555555555556 \cdot {x}^{2}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{{x}^{2} \cdot -0.005555555555555556}\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot -0.005555555555555556\right)} \]

   if 1.0000100000000001 < (/.f64 (sinh.f64 x) x)

   1. Initial program 76.7%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 75.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{\sinh x}{x}\right)\right)\right)} \]

      2. expm1-undefine 75.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{\sinh x}{x}\right)\right)} - 1} \]

      3. log1p-undefine 75.2%

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

      4. rem-exp-log 76.1%

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

   4. Applied egg-rr 76.1%

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

      1. associate--l+ 77.2%

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

      2. +-commutative 77.2%

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

      3. sub-neg 77.2%

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

      4. metadata-eval 77.2%

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

   6. Applied egg-rr 77.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.00001:\\ \;\;\;\;{x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot -0.005555555555555556\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{\sinh x}{x}\right) + -1\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sinh x\_m}{x\_m}\\ \mathbf{if}\;t\_0 \leq 1.0000001:\\ \;\;\;\;{x\_m}^{2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(\log t\_0 + -1\right) + 1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ (sinh x_m) x_m)))
   (if (<= t_0 1.0000001)
     (* (pow x_m 2.0) 0.16666666666666666)
     (+ (+ (log t_0) -1.0) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sinh(x_m) / x_m;
	double tmp;
	if (t_0 <= 1.0000001) {
		tmp = pow(x_m, 2.0) * 0.16666666666666666;
	} else {
		tmp = (log(t_0) + -1.0) + 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(x_m) / x_m
    if (t_0 <= 1.0000001d0) then
        tmp = (x_m ** 2.0d0) * 0.16666666666666666d0
    else
        tmp = (log(t_0) + (-1.0d0)) + 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sinh(x_m) / x_m;
	double tmp;
	if (t_0 <= 1.0000001) {
		tmp = Math.pow(x_m, 2.0) * 0.16666666666666666;
	} else {
		tmp = (Math.log(t_0) + -1.0) + 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sinh(x_m) / x_m
	tmp = 0
	if t_0 <= 1.0000001:
		tmp = math.pow(x_m, 2.0) * 0.16666666666666666
	else:
		tmp = (math.log(t_0) + -1.0) + 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(sinh(x_m) / x_m)
	tmp = 0.0
	if (t_0 <= 1.0000001)
		tmp = Float64((x_m ^ 2.0) * 0.16666666666666666);
	else
		tmp = Float64(Float64(log(t_0) + -1.0) + 1.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = sinh(x_m) / x_m;
	tmp = 0.0;
	if (t_0 <= 1.0000001)
		tmp = (x_m ^ 2.0) * 0.16666666666666666;
	else
		tmp = (log(t_0) + -1.0) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Sinh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 1.0000001], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[(N[(N[Log[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 x) x) < 1.00000010000000006

   1. Initial program 50.9%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2}} \]

   if 1.00000010000000006 < (/.f64 (sinh.f64 x) x)

   1. Initial program 75.8%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. expm1-log1p-u 74.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{\sinh x}{x}\right)\right)\right)} \]

      2. expm1-undefine 74.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(\frac{\sinh x}{x}\right)\right)} - 1} \]

      3. log1p-undefine 74.7%

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

      4. rem-exp-log 75.5%

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

   4. Applied egg-rr 75.5%

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

      1. associate--l+ 76.5%

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

      2. +-commutative 76.5%

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

      3. sub-neg 76.5%

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

      4. metadata-eval 76.5%

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

   6. Applied egg-rr 76.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.0000001:\\ \;\;\;\;{x}^{2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{\sinh x}{x}\right) + -1\right) + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\sinh x\_m}{x\_m}\\ \mathbf{if}\;t\_0 \leq 1.0000001:\\ \;\;\;\;{x\_m}^{2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log t\_0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ (sinh x_m) x_m)))
   (if (<= t_0 1.0000001) (* (pow x_m 2.0) 0.16666666666666666) (log t_0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sinh(x_m) / x_m;
	double tmp;
	if (t_0 <= 1.0000001) {
		tmp = pow(x_m, 2.0) * 0.16666666666666666;
	} else {
		tmp = log(t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(x_m) / x_m
    if (t_0 <= 1.0000001d0) then
        tmp = (x_m ** 2.0d0) * 0.16666666666666666d0
    else
        tmp = log(t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.sinh(x_m) / x_m;
	double tmp;
	if (t_0 <= 1.0000001) {
		tmp = Math.pow(x_m, 2.0) * 0.16666666666666666;
	} else {
		tmp = Math.log(t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.sinh(x_m) / x_m
	tmp = 0
	if t_0 <= 1.0000001:
		tmp = math.pow(x_m, 2.0) * 0.16666666666666666
	else:
		tmp = math.log(t_0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(sinh(x_m) / x_m)
	tmp = 0.0
	if (t_0 <= 1.0000001)
		tmp = Float64((x_m ^ 2.0) * 0.16666666666666666);
	else
		tmp = log(t_0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = sinh(x_m) / x_m;
	tmp = 0.0;
	if (t_0 <= 1.0000001)
		tmp = (x_m ^ 2.0) * 0.16666666666666666;
	else
		tmp = log(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Sinh[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 1.0000001], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision], N[Log[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 x) x) < 1.00000010000000006

   1. Initial program 50.9%

      \[\log \left(\frac{\sinh x}{x}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2}} \]

   if 1.00000010000000006 < (/.f64 (sinh.f64 x) x)

   1. Initial program 75.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh x}{x} \leq 1.0000001:\\ \;\;\;\;{x}^{2} \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sinh x}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {x\_m}^{2} \cdot 0.16666666666666666 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow x_m 2.0) 0.16666666666666666))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(x_m, 2.0) * 0.16666666666666666;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m ** 2.0d0) * 0.16666666666666666d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(x_m, 2.0) * 0.16666666666666666;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(x_m, 2.0) * 0.16666666666666666

Julia

x_m = abs(x)
function code(x_m)
	return Float64((x_m ^ 2.0) * 0.16666666666666666)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m ^ 2.0) * 0.16666666666666666;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 51.9%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 96.4%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2}} \]

4. Final simplification 96.4%

   \[\leadsto {x}^{2} \cdot 0.16666666666666666 \]
5. Add Preprocessing

Alternative 6: 50.5% accurate, 203.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 51.9%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 48.5%

   \[\leadsto \log \color{blue}{1} \]

4. Final simplification 48.5%

   \[\leadsto 0 \]
5. Add Preprocessing

Developer target: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "bug500, discussion (missed optimization)"
  :precision binary64

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

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