VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 7.6s

Alternatives: 9

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	return -(x * (1.0 / tan(B))) + (1.0 / sin(B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
end function

Java

public static double code(double B, double x) {
	return -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
}

Python

def code(B, x):
	return -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))

Julia

function code(B, x)
	return Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
end

Wolfram

code[B_, x_] := N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{\sin B} \cdot \cos B + {\sin B}^{-1} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (* (/ (- x) (sin B)) (cos B)) (pow (sin B) -1.0)))

C

double code(double B, double x) {
	return ((-x / sin(B)) * cos(B)) + pow(sin(B), -1.0);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((-x / sin(b)) * cos(b)) + (sin(b) ** (-1.0d0))
end function

Java

public static double code(double B, double x) {
	return ((-x / Math.sin(B)) * Math.cos(B)) + Math.pow(Math.sin(B), -1.0);
}

Python

def code(B, x):
	return ((-x / math.sin(B)) * math.cos(B)) + math.pow(math.sin(B), -1.0)

Julia

function code(B, x)
	return Float64(Float64(Float64(Float64(-x) / sin(B)) * cos(B)) + (sin(B) ^ -1.0))
end

MATLAB

function tmp = code(B, x)
	tmp = ((-x / sin(B)) * cos(B)) + (sin(B) ^ -1.0);
end

Wolfram

code[B_, x_] := N[(N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \frac{-x}{\sin B} \cdot \cos B + {\sin B}^{-1} \]
6. Add Preprocessing

Alternative 2: 98.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \mathbf{if}\;t\_0 \leq -10 \lor \neg \left(t\_0 \leq 1000000\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (pow (sin B) -1.0))))
   (if (or (<= t_0 -10.0) (not (<= t_0 1000000.0)))
     (+ (/ (- x) (tan B)) (pow B -1.0))
     (/ (- 1.0 x) (sin B)))))

C

double code(double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + pow(sin(B), -1.0);
	double tmp;
	if ((t_0 <= -10.0) || !(t_0 <= 1000000.0)) {
		tmp = (-x / tan(B)) + pow(B, -1.0);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) + (sin(b) ** (-1.0d0))
    if ((t_0 <= (-10.0d0)) .or. (.not. (t_0 <= 1000000.0d0))) then
        tmp = (-x / tan(b)) + (b ** (-1.0d0))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) + Math.pow(Math.sin(B), -1.0);
	double tmp;
	if ((t_0 <= -10.0) || !(t_0 <= 1000000.0)) {
		tmp = (-x / Math.tan(B)) + Math.pow(B, -1.0);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + math.pow(math.sin(B), -1.0)
	tmp = 0
	if (t_0 <= -10.0) or not (t_0 <= 1000000.0):
		tmp = (-x / math.tan(B)) + math.pow(B, -1.0)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + (sin(B) ^ -1.0))
	tmp = 0.0
	if ((t_0 <= -10.0) || !(t_0 <= 1000000.0))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + (B ^ -1.0));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (x * (-1.0 / tan(B))) + (sin(B) ^ -1.0);
	tmp = 0.0;
	if ((t_0 <= -10.0) || ~((t_0 <= 1000000.0)))
		tmp = (-x / tan(B)) + (B ^ -1.0);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -10.0], N[Not[LessEqual[t$95$0, 1000000.0]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -10 or 1e6 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B)))

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\sin B} \cdot \cos B\right)\right)} + \frac{1}{\sin B} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\sin B} \cdot \cos B}\right)\right) + \frac{1}{\sin B} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\sin B}} \cdot \cos B\right)\right) + \frac{1}{\sin B} \]

      4. associate-*l/ N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) + \frac{1}{\sin B} \]

      5. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot \cos B\right)}{\sin B}} + \frac{1}{\sin B} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \cos B}}{\sin B} + \frac{1}{\sin B} \]

      7. lift-neg.f64 N/A

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

      8. associate-*r/ N/A

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

      9. clear-num N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-tan.f64 N/A

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

      14. un-div-inv N/A

          \[\leadsto \color{blue}{\frac{-x}{\tan B}} + \frac{1}{\sin B} \]

      15. lower-/.f64 99.8

          \[\leadsto \color{blue}{\frac{-x}{\tan B}} + \frac{1}{\sin B} \]

   6. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} + \frac{1}{\sin B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{-x}{\tan B} + \color{blue}{\frac{1}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 99.1

         \[\leadsto \frac{-x}{\tan B} + \color{blue}{\frac{1}{B}} \]

   9. Applied rewrites 99.1%

      \[\leadsto \frac{-x}{\tan B} + \color{blue}{\frac{1}{B}} \]

   if -10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < 1e6

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\sin B} \cdot \cos B\right)\right)} \]

      4. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B} \]

      5. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\sin B} \cdot \cos B \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

      8. associate-*l/ N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      9. sub-div N/A

         \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{1 - x \cdot \cos B}}{\sin B} \]

      12. *-commutative N/A

          \[\leadsto \frac{1 - \color{blue}{\cos B \cdot x}}{\sin B} \]

      13. lower-*.f64 99.6

          \[\leadsto \frac{1 - \color{blue}{\cos B \cdot x}}{\sin B} \]

   6. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]
   8. Step-by-step derivation

      1. lower--.f64 96.1

         \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]

   9. Applied rewrites 96.1%

      \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq -10 \lor \neg \left(x \cdot \frac{-1}{\tan B} + {\sin B}^{-1} \leq 1000000\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{\tan B} + {\sin B}^{-1} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (+ (/ (- x) (tan B)) (pow (sin B) -1.0)))

C

double code(double B, double x) {
	return (-x / tan(B)) + pow(sin(B), -1.0);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-x / tan(b)) + (sin(b) ** (-1.0d0))
end function

Java

public static double code(double B, double x) {
	return (-x / Math.tan(B)) + Math.pow(Math.sin(B), -1.0);
}

Python

def code(B, x):
	return (-x / math.tan(B)) + math.pow(math.sin(B), -1.0)

Julia

function code(B, x)
	return Float64(Float64(Float64(-x) / tan(B)) + (sin(B) ^ -1.0))
end

MATLAB

function tmp = code(B, x)
	tmp = (-x / tan(B)) + (sin(B) ^ -1.0);
end

Wolfram

code[B_, x_] := N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \frac{-x}{\tan B} + {\sin B}^{-1} \]
6. Add Preprocessing

Alternative 4: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos B \cdot x}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 (* (cos B) x)) (sin B)))

C

double code(double B, double x) {
	return (1.0 - (cos(B) * x)) / sin(B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - (cos(b) * x)) / sin(b)
end function

Java

public static double code(double B, double x) {
	return (1.0 - (Math.cos(B) * x)) / Math.sin(B);
}

Python

def code(B, x):
	return (1.0 - (math.cos(B) * x)) / math.sin(B)

Julia

function code(B, x)
	return Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - (cos(B) * x)) / sin(B);
end

Wolfram

code[B_, x_] := N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

      \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\sin B} \cdot \cos B\right)\right)} \]

   4. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B} \]

   5. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\sin B} \cdot \cos B \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

   8. associate-*l/ N/A

      \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

   9. sub-div N/A

      \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\color{blue}{1 - x \cdot \cos B}}{\sin B} \]

   12. *-commutative N/A

       \[\leadsto \frac{1 - \color{blue}{\cos B \cdot x}}{\sin B} \]

   13. lower-*.f64 99.7

       \[\leadsto \frac{1 - \color{blue}{\cos B \cdot x}}{\sin B} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]
7. Add Preprocessing

Alternative 5: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) (sin B)))

C

double code(double B, double x) {
	return (1.0 - x) / sin(B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / sin(b)
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / Math.sin(B);
}

Python

def code(B, x):
	return (1.0 - x) / math.sin(B)

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / sin(B))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / sin(B);
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lift-tan.f64 N/A

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

   5. tan-quot N/A

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

   6. lift-sin.f64 N/A

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

   7. associate-/r/ N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

      \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{\sin B} \cdot \cos B\right)\right)} \]

   4. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B} \]

   5. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\sin B} \cdot \cos B \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{\sin B}} \cdot \cos B \]

   8. associate-*l/ N/A

      \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

   9. sub-div N/A

      \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1 - x \cdot \cos B}{\sin B}} \]

   11. lower--.f64 N/A

       \[\leadsto \frac{\color{blue}{1 - x \cdot \cos B}}{\sin B} \]

   12. *-commutative N/A

       \[\leadsto \frac{1 - \color{blue}{\cos B \cdot x}}{\sin B} \]

   13. lower-*.f64 99.7

       \[\leadsto \frac{1 - \color{blue}{\cos B \cdot x}}{\sin B} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]

7. Taylor expanded in B around 0

   \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]
8. Step-by-step derivation

   1. lower--.f64 78.3

      \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]

9. Applied rewrites 78.3%

   \[\leadsto \frac{\color{blue}{1 - x}}{\sin B} \]
10. Add Preprocessing

Alternative 6: 51.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (/ (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x) B))

C

double code(double B, double x) {
	return (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
}

Julia

function code(B, x)
	return Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B)
end

Wolfram

code[B_, x_] := N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}}{B} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right) + 1\right)} - x}{B} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{3} \cdot x\right) \cdot {B}^{2}} + 1\right) - x}{B} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{3} \cdot x, {B}^{2}, 1\right)} - x}{B} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{3} \cdot x + \frac{1}{6}}, {B}^{2}, 1\right) - x}{B} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right)}, {B}^{2}, 1\right) - x}{B} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right), \color{blue}{B \cdot B}, 1\right) - x}{B} \]

   9. lower-*.f64 53.1

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), \color{blue}{B \cdot B}, 1\right) - x}{B} \]

5. Applied rewrites 53.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}} \]
6. Add Preprocessing

Alternative 7: 51.2% accurate, 15.5× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) B))

C

double code(double B, double x) {
	return (1.0 - x) / B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / b
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / B;
}

Python

def code(B, x):
	return (1.0 - x) / B

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / B)
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / B;
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   2. lower--.f64 52.8

      \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
6. Add Preprocessing

Alternative 8: 27.0% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ 1.0 B))

C

double code(double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double B, double x) {
	return 1.0 / B;
}

Python

def code(B, x):
	return 1.0 / B

Julia

function code(B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(B, x)
	tmp = 1.0 / B;
end

Wolfram

code[B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

4. Applied rewrites 44.0%

   \[\leadsto \color{blue}{\frac{\sin B + \frac{\tan B}{x}}{\frac{\tan B}{x} \cdot \sin B}} \]

5. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. rgt-mult-inverse N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f64 24.3

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

7. Applied rewrites 24.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
9. Step-by-step derivation

10. Applied rewrites 24.9%

    \[\leadsto \frac{1}{B} \]
11. Add Preprocessing

Alternative 9: 2.5% accurate, 19.4× speedup

Math

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

FPCore

(FPCore (B x) :precision binary64 (/ x B))

C

double code(double B, double x) {
	return x / B;
}

Fortran

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

Java

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

Python

def code(B, x):
	return x / B

Julia

function code(B, x)
	return Float64(x / B)
end

MATLAB

function tmp = code(B, x)
	tmp = x / B;
end

Wolfram

code[B_, x_] := N[(x / B), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. Add Preprocessing

4. Applied rewrites 44.0%

   \[\leadsto \color{blue}{\frac{\sin B + \frac{\tan B}{x}}{\frac{\tan B}{x} \cdot \sin B}} \]

5. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. rgt-mult-inverse N/A

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

   5. *-rgt-identity N/A

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

   6. lower-+.f64 24.3

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

7. Applied rewrites 24.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
9. Step-by-step derivation

10. Applied rewrites 24.9%

    \[\leadsto \frac{1}{B} \]

11. Taylor expanded in x around inf

    \[\leadsto \frac{x}{\color{blue}{B}} \]
12. Step-by-step derivation

13. Applied rewrites 2.3%

    \[\leadsto \frac{x}{\color{blue}{B}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))