VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 10.5s

Alternatives: 13

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\tan B}, -x, \frac{1}{\sin B}\right)} \]
5. Step-by-step derivation

   1. tan-quot N/A

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

   2. lift-sin.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 99.7

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

6. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\cos B}{\sin B}}, -x, \frac{1}{\sin B}\right) \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. unsub-neg N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. associate-*l/ N/A

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

   14. sub-div N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-*.f64 99.7

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0048:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -1.6) t_0 (if (<= x 0.0048) (/ (- 1.0 x) (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -1.6) {
		tmp = t_0;
	} else if (x <= 0.0048) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = t_0;
	}
	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 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-1.6d0)) then
        tmp = t_0
    else if (x <= 0.0048d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -1.6) {
		tmp = t_0;
	} else if (x <= 0.0048) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -1.6:
		tmp = t_0
	elif x <= 0.0048:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.6)
		tmp = t_0;
	elseif (x <= 0.0048)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -1.6)
		tmp = t_0;
	elseif (x <= 0.0048)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6], t$95$0, If[LessEqual[x, 0.0048], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 0.00479999999999999958 < x

   1. Initial program 99.6%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 98.6

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

   7. Simplified 98.6%

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

   if -1.6000000000000001 < x < 0.00479999999999999958

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\tan B}, -x, \frac{1}{\sin B}\right)} \]
   5. Step-by-step derivation

      1. tan-quot N/A

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

      2. lift-sin.f64 N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.8

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

   6. Applied egg-rr 99.8%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\cos B}{\sin B}}, -x, \frac{1}{\sin B}\right) \]
   7. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. unsub-neg N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

      14. sub-div N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 99.8

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in B around 0

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

       1. lower--.f64 98.8

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

   11. Simplified 98.8%

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

Alternative 4: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.49:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), 0.019444444444444445\right)\right), \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)\right), 1\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.49)
   (/
    (-
     (fma
      (* B B)
      (fma
       B
       (*
        B
        (fma
         x
         0.022222222222222223
         (fma
          (* B B)
          (fma x 0.0021164021164021165 0.00205026455026455)
          0.019444444444444445)))
       (fma x 0.3333333333333333 0.16666666666666666))
      1.0)
     x)
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.49) {
		tmp = (fma((B * B), fma(B, (B * fma(x, 0.022222222222222223, fma((B * B), fma(x, 0.0021164021164021165, 0.00205026455026455), 0.019444444444444445))), fma(x, 0.3333333333333333, 0.16666666666666666)), 1.0) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.49)
		tmp = Float64(Float64(fma(Float64(B * B), fma(B, Float64(B * fma(x, 0.022222222222222223, fma(Float64(B * B), fma(x, 0.0021164021164021165, 0.00205026455026455), 0.019444444444444445))), fma(x, 0.3333333333333333, 0.16666666666666666)), 1.0) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.49], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(B * N[(B * N[(x * 0.022222222222222223 + N[(N[(B * B), $MachinePrecision] * N[(x * 0.0021164021164021165 + 0.00205026455026455), $MachinePrecision] + 0.019444444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 0.3333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.48999999999999999

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{31}{15120} + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B, B \cdot \mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(x, 0.0021164021164021165, 0.00205026455026455\right), 0.019444444444444445\right)\right), \mathsf{fma}\left(x, 0.3333333333333333, 0.16666666666666666\right)\right), 1\right) - x}{B}} \]

   if 0.48999999999999999 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 46.6

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

   5. Simplified 46.6%

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

Alternative 5: 77.2% 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 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\tan B}, -x, \frac{1}{\sin B}\right)} \]
5. Step-by-step derivation

   1. tan-quot N/A

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

   2. lift-sin.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lower-cos.f64 99.7

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

6. Applied egg-rr 99.7%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\cos B}{\sin B}}, -x, \frac{1}{\sin B}\right) \]
7. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. unsub-neg N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. associate-*l/ N/A

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

   14. sub-div N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-*.f64 99.7

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in B around 0

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

    1. lower--.f64 77.0

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

11. Simplified 77.0%

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

Alternative 6: 51.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), B \cdot \left(B \cdot B\right), B\right)} - \frac{x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (-
  (/
   1.0
   (fma
    (fma
     (* B B)
     (fma (* B B) -0.0001984126984126984 0.008333333333333333)
     -0.16666666666666666)
    (* B (* B B))
    B))
  (/ x B)))

C

double code(double B, double x) {
	return (1.0 / fma(fma((B * B), fma((B * B), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), (B * (B * B)), B)) - (x / B);
}

Julia

function code(B, x)
	return Float64(Float64(1.0 / fma(fma(Float64(B * B), fma(Float64(B * B), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), Float64(B * Float64(B * B)), B)) - Float64(x / B))
end

Wolfram

code[B_, x_] := N[(N[(1.0 / N[(N[(N[(B * B), $MachinePrecision] * N[(N[(B * B), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(B * N[(B * B), $MachinePrecision]), $MachinePrecision] + B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 75.2

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

5. Simplified 75.2%

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

6. Taylor expanded in B around 0

   \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{\color{blue}{B \cdot \left(1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right)}} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{B \cdot \color{blue}{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right) + 1\right)}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{\color{blue}{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right)\right) \cdot B + 1 \cdot B}} \]

   3. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{\color{blue}{\left(\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right) \cdot {B}^{2}\right)} \cdot B + 1 \cdot B} \]

   4. associate-*l* N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{\color{blue}{\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right) \cdot \left({B}^{2} \cdot B\right)} + 1 \cdot B} \]

   5. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(B \cdot B\right)} \cdot B\right) + 1 \cdot B} \]

   6. unpow3 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{B}^{3}} + 1 \cdot B} \]

   7. *-lft-identity N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}\right) \cdot {B}^{3} + \color{blue}{B}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{1}{\color{blue}{\mathsf{fma}\left({B}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {B}^{2}\right) - \frac{1}{6}, {B}^{3}, B\right)}} \]

8. Simplified 51.3%

   \[\leadsto \left(-\frac{x}{B}\right) + \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), B \cdot \left(B \cdot B\right), B\right)}} \]

9. Final simplification 51.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(B \cdot B, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), B \cdot \left(B \cdot B\right), B\right)} - \frac{x}{B} \]
10. Add Preprocessing

Alternative 7: 51.4% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 75.2

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

5. Simplified 75.2%

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

6. Taylor expanded in B around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 51.2

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

8. Simplified 51.2%

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

9. Final simplification 51.2%

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

Alternative 8: 51.5% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 51.2

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

5. Simplified 51.2%

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

Alternative 9: 50.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= x -2.4e-8) t_0 (if (<= x 1.0) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -2.4e-8) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	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 / b)
    if (x <= (-2.4d-8)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -2.4e-8) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -(x / B)
	tmp = 0
	if x <= -2.4e-8:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (x <= -2.4e-8)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (x <= -2.4e-8)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[x, -2.4e-8], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999998e-8 or 1 < x

   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 49.1

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

   5. Simplified 49.1%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{B} \]

      2. lower-neg.f64 48.9

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

   8. Simplified 48.9%

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

   if -2.39999999999999998e-8 < x < 1

   1. Initial program 99.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 53.2

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

   5. Simplified 53.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 51.2

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

   8. Simplified 51.2%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-8}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.4% 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 51.2

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

5. Simplified 51.2%

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

Alternative 11: 26.4% 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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 51.2

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

5. Simplified 51.2%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 27.5

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

8. Simplified 27.5%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
9. Add Preprocessing

Alternative 12: 3.1% accurate, 38.8× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.5 \end{array} \]

FPCore

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

C

double code(double B, double x) {
	return B * 0.5;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = b * 0.5d0
end function

Java

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

Python

def code(B, x):
	return B * 0.5

Julia

function code(B, x)
	return Float64(B * 0.5)
end

MATLAB

function tmp = code(B, x)
	tmp = B * 0.5;
end

Wolfram

code[B_, x_] := N[(B * 0.5), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot B} \]
6. Step-by-step derivation

   1. lower-*.f64 3.1

      \[\leadsto \color{blue}{0.5 \cdot B} \]

7. Simplified 3.1%

   \[\leadsto \color{blue}{0.5 \cdot B} \]

8. Final simplification 3.1%

   \[\leadsto B \cdot 0.5 \]
9. Add Preprocessing

Alternative 13: 2.6% accurate, 77.7× speedup

Math

\[\begin{array}{l} \\ -B \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(B, x):
	return -B

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := (-B)

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot B} \]
6. Step-by-step derivation

   1. lower-*.f64 3.1

      \[\leadsto \color{blue}{0.5 \cdot B} \]

7. Simplified 3.1%

   \[\leadsto \color{blue}{0.5 \cdot B} \]

8. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot B} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(B\right)} \]

   2. lower-neg.f64 2.6

      \[\leadsto \color{blue}{-B} \]

10. Simplified 2.6%

    \[\leadsto \color{blue}{-B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))