Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.5s

Alternatives: 7

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "2.581602127188603564692636185742382587966e125")
  2. reg1 = (bf #e80560.27817607086035422980785369873046875)

• 48.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x}{\frac{y}{\sinh y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (sin x) (/ y (sinh y))))

C

double code(double x, double y) {
	return sin(x) / (y / sinh(y));
}

Fortran

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

Java

public static double code(double x, double y) {
	return Math.sin(x) / (y / Math.sinh(y));
}

Python

def code(x, y):
	return math.sin(x) / (y / math.sinh(y))

Julia

function code(x, y)
	return Float64(sin(x) / Float64(y / sinh(y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) / (y / sinh(y));
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 77.9%

      \[\leadsto \color{blue}{\log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

   2. *-un-lft-identity 77.9%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\sin x \cdot \frac{\sinh y}{y}}\right)} \]

   3. log-prod 77.9%

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

   4. metadata-eval 77.9%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

   2. associate-*r/ 89.9%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sinh y}{y}} \]

   3. associate-*l/ 91.2%

      \[\leadsto \color{blue}{\frac{\sin x}{y} \cdot \sinh y} \]

   4. associate-/r/ 100.0%

      \[\leadsto \color{blue}{\frac{\sin x}{\frac{y}{\sinh y}}} \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\sin x}{\frac{y}{\sinh y}}} \]
7. Add Preprocessing

Alternative 2: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 2.0) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if (t_0 <= 2.0d0) then
        tmp = sin(x)
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 2.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 2.0:
		tmp = math.sin(x)
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = sin(x);
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 2

   1. Initial program 99.9%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.0%

      \[\leadsto \color{blue}{\sin x} \]

   if 2 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 71.3%

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

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 4: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1000000:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1000000.0) (sin x) (* x (+ 1.0 (* (* x x) -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1000000.0) {
		tmp = sin(x);
	} else {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1000000.0d0) then
        tmp = sin(x)
    else
        tmp = x * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1000000.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1000000.0:
		tmp = math.sin(x)
	else:
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1000000.0)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1000000.0)
		tmp = sin(x);
	else
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1000000.0], N[Sin[x], $MachinePrecision], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1e6

   1. Initial program 99.9%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 68.7%

      \[\leadsto \color{blue}{\sin x} \]

   if 1e6 < y

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 2.7%

      \[\leadsto \color{blue}{\sin x} \]

   4. Taylor expanded in x around 0 21.3%

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

      1. *-commutative 21.3%

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

   6. Simplified 21.3%

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

      1. unpow2 21.3%

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

   8. Applied egg-rr 21.3%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 34.6% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (+ 1.0 (* (* x x) -0.16666666666666666))))

C

double code(double x, double y) {
	return x * (1.0 + ((x * x) * -0.16666666666666666));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 55.9%

   \[\leadsto \color{blue}{\sin x} \]

4. Taylor expanded in x around 0 33.4%

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

   1. *-commutative 33.4%

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

6. Simplified 33.4%

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

   1. unpow2 33.4%

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

8. Applied egg-rr 33.4%

   \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.16666666666666666\right) \]
9. Add Preprocessing

Alternative 6: 26.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

double code(double x, double y) {
	return x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x
end function

Java

public static double code(double x, double y) {
	return x;
}

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

function tmp = code(x, y)
	tmp = x;
end

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 56.0%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]

4. Taylor expanded in y around 0 25.1%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Alternative 7: 3.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 100.0%

   \[\sin x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 99.9%

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

   2. expm1-undefine 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \sin x \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sinh y}{y}\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-define 99.9%

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

6. Simplified 99.9%

   \[\leadsto \sin x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sinh y}{y}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-undefine 100.0%

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

   2. log1p-undefine 100.0%

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

   3. rem-exp-log 100.0%

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

   4. +-commutative 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \sin x \cdot \color{blue}{\left(\left(\frac{\sinh y}{y} + 1\right) - 1\right)} \]
9. Step-by-step derivation

   1. sub-neg 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. metadata-eval 100.0%

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

10. Applied egg-rr 100.0%

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

11. Taylor expanded in y around inf 2.9%

    \[\leadsto \color{blue}{\sin x + -1 \cdot \sin x} \]
12. Step-by-step derivation

    1. distribute-rgt1-in 2.9%

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

    2. metadata-eval 2.9%

       \[\leadsto \color{blue}{0} \cdot \sin x \]

    3. mul0-lft 2.9%

       \[\leadsto \color{blue}{0} \]

13. Simplified 2.9%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))