VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.0s

Alternatives: 11

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
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 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := t\_0 - \frac{1}{\tan B} \cdot x\\ t_2 := {B}^{-1} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000000:\\ \;\;\;\;t\_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1 (- t_0 (* (/ 1.0 (tan B)) x)))
        (t_2 (- (pow B -1.0) (/ x (tan B)))))
   (if (<= t_1 -1000000.0) t_2 (if (<= t_1 2000000.0) (- t_0 (/ x B)) t_2))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = t_0 - ((1.0 / tan(B)) * x);
	double t_2 = pow(B, -1.0) - (x / tan(B));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2000000.0) {
		tmp = t_0 - (x / B);
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = t_0 - ((1.0d0 / tan(b)) * x)
    t_2 = (b ** (-1.0d0)) - (x / tan(b))
    if (t_1 <= (-1000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 2000000.0d0) then
        tmp = t_0 - (x / b)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = t_0 - ((1.0 / Math.tan(B)) * x);
	double t_2 = Math.pow(B, -1.0) - (x / Math.tan(B));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2000000.0) {
		tmp = t_0 - (x / B);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = t_0 - ((1.0 / math.tan(B)) * x)
	t_2 = math.pow(B, -1.0) - (x / math.tan(B))
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = t_2
	elif t_1 <= 2000000.0:
		tmp = t_0 - (x / B)
	else:
		tmp = t_2
	return tmp

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(t_0 - Float64(Float64(1.0 / tan(B)) * x))
	t_2 = Float64((B ^ -1.0) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 2000000.0)
		tmp = Float64(t_0 - Float64(x / B));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 1.0 / sin(B);
	t_1 = t_0 - ((1.0 / tan(B)) * x);
	t_2 = (B ^ -1.0) - (x / tan(B));
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 2000000.0)
		tmp = t_0 - (x / B);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, -1.0], $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, 2000000.0], N[(t$95$0 - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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))) < -1e6 or 2e6 < (+.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. 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. rem-log-exp N/A

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

      6. pow-to-exp N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + e^{\color{blue}{\mathsf{neg}\left(\log \sin B\right)}} \]

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 51.1

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

   6. Applied rewrites 51.1%

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

   7. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{e^{\mathsf{neg}\left(\log B\right)}} \]
   8. Step-by-step derivation

      1. neg-mul-1 N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. lower-pow.f64 99.5

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

   9. Applied rewrites 99.5%

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

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

   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

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

      1. lower-/.f64 95.2

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

   5. Applied rewrites 95.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} - \frac{1}{\tan B} \cdot x \leq -1000000:\\ \;\;\;\;{B}^{-1} - \frac{x}{\tan B}\\ \mathbf{elif}\;\frac{1}{\sin B} - \frac{1}{\tan B} \cdot x \leq 2000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;{B}^{-1} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. lift-tan.f64 N/A

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

   10. tan-quot N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. clear-num N/A

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

   14. associate-*l/ N/A

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

   15. sub-div N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 N/A

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

   18. 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 4: 83.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ (fma 0.16666666666666666 (* B B) 1.0) B) (/ x (tan B)))))
   (if (<= x -1.15e+71) t_0 (if (<= x 1.95) (/ (- 1.0 x) (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (fma(0.16666666666666666, (B * B), 1.0) / B) - (x / tan(B));
	double tmp;
	if (x <= -1.15e+71) {
		tmp = t_0;
	} else if (x <= 1.95) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(B, x)
	t_0 = Float64(Float64(fma(0.16666666666666666, Float64(B * B), 1.0) / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.15e+71)
		tmp = t_0;
	elseif (x <= 1.95)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(N[(0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e+71], t$95$0, If[LessEqual[x, 1.95], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1500000000000001e71 or 1.94999999999999996 < x

   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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

   if -1.1500000000000001e71 < x < 1.94999999999999996

   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. 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lift-tan.f64 N/A

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

      10. tan-quot N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. clear-num N/A

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

      14. associate-*l/ N/A

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

      15. sub-div N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. lower--.f64 95.7

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

   9. Applied rewrites 95.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.0% 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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. lower-/.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-neg.f64 N/A

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

   4. unsub-neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. *-lft-identity N/A

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

   8. associate-*l/ N/A

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

   9. lift-tan.f64 N/A

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

   10. tan-quot N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-cos.f64 N/A

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

   13. clear-num N/A

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

   14. associate-*l/ N/A

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

   15. sub-div N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 N/A

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

   18. 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 75.2

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

9. Applied rewrites 75.2%

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

Alternative 6: 50.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(B \cdot x\right) \cdot 0.022222222222222223, B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (/
  (fma
   (fma
    (* (* B x) 0.022222222222222223)
    B
    (fma 0.3333333333333333 x 0.16666666666666666))
   (* B B)
   (- 1.0 x))
  B))

C

double code(double B, double x) {
	return fma(fma(((B * x) * 0.022222222222222223), B, fma(0.3333333333333333, x, 0.16666666666666666)), (B * B), (1.0 - x)) / B;
}

Julia

function code(B, x)
	return Float64(fma(fma(Float64(Float64(B * x) * 0.022222222222222223), B, fma(0.3333333333333333, x, 0.16666666666666666)), Float64(B * B), Float64(1.0 - x)) / B)
end

Wolfram

code[B_, x_] := N[(N[(N[(N[(N[(B * x), $MachinePrecision] * 0.022222222222222223), $MachinePrecision] * B + N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $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. 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 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{7}{360} + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

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

7. Applied rewrites 48.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{45} \cdot \left(B \cdot x\right), B, \mathsf{fma}\left(\frac{1}{3}, x, \frac{1}{6}\right)\right), B \cdot B, 1 - x\right)}{B} \]
9. Step-by-step derivation

10. Applied rewrites 48.5%

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

11. Final simplification 48.5%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(B \cdot x\right) \cdot 0.022222222222222223, B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right)\right), B \cdot B, 1 - x\right)}{B} \]
12. Add Preprocessing

Alternative 7: 50.5% 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 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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. *-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 48.5

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

5. Applied rewrites 48.5%

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

Alternative 8: 50.5% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[B_, x_] := N[(N[(N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] * x + 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. 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 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. rem-log-exp N/A

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

   6. pow-to-exp N/A

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

   7. inv-pow N/A

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

   8. log-rec N/A

      \[\leadsto \left(-\frac{x}{\tan B}\right) + e^{\color{blue}{\mathsf{neg}\left(\log \sin B\right)}} \]

   9. lower-neg.f64 N/A

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

   10. lower-log.f64 52.6

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

6. Applied rewrites 52.6%

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

7. Taylor expanded in B around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 23.4

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

9. Applied rewrites 23.4%

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

10. Taylor expanded in B around 0

    \[\leadsto \color{blue}{\frac{B \cdot \left(e^{\mathsf{neg}\left(\log B\right)} + \frac{1}{3} \cdot \left(B \cdot x\right)\right) - x}{B}} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{B \cdot \left(e^{\mathsf{neg}\left(\log B\right)} + \frac{1}{3} \cdot \left(B \cdot x\right)\right) - x}{B}} \]

12. Applied rewrites 48.3%

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

13. Final simplification 48.3%

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

Alternative 9: 49.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B))) (if (<= x -1.0) t_0 (if (<= x 2.0) (/ 1.0 B) t_0))))

C

double code(double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		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 <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 2.0d0) 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 <= -1.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -1.0:
		tmp = t_0
	elif x <= 2.0:
		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 <= -1.0)
		tmp = t_0;
	elseif (x <= 2.0)
		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 <= -1.0)
		tmp = t_0;
	elseif (x <= 2.0)
		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, -1.0], t$95$0, If[LessEqual[x, 2.0], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 2 < x

   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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. rem-log-exp N/A

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

      6. pow-to-exp N/A

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

      7. inv-pow N/A

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

      8. log-rec N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + e^{\color{blue}{\mathsf{neg}\left(\log \sin B\right)}} \]

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 53.9

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

   6. Applied rewrites 53.9%

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

   7. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 45.7

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

   9. Applied rewrites 45.7%

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

   if -1 < x < 2

   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 49.3

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

   5. Applied rewrites 49.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.9%

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

Alternative 10: 50.4% 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 48.1

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

5. Applied rewrites 48.1%

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

Alternative 11: 25.4% 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 48.1

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

5. Applied rewrites 48.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2024295 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))