VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.8s

Alternatives: 12

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 rewrites 99.7%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ t_2 := t\_0 + t\_1\\ \mathbf{if}\;t\_2 \leq -20000:\\ \;\;\;\;\frac{1}{B} + \frac{\frac{1}{\tan B}}{\frac{-1}{x}}\\ \mathbf{elif}\;t\_2 \leq 2000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (* x (/ -1.0 (tan B)))) (t_2 (+ t_0 t_1)))
   (if (<= t_2 -20000.0)
     (+ (/ 1.0 B) (/ (/ 1.0 (tan B)) (/ -1.0 x)))
     (if (<= t_2 2000000.0) t_0 (+ t_1 (/ 1.0 B))))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x * (-1.0 / tan(B));
	double t_2 = t_0 + t_1;
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = (1.0 / B) + ((1.0 / tan(B)) / (-1.0 / x));
	} else if (t_2 <= 2000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (1.0 / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x * ((-1.0d0) / tan(b))
    t_2 = t_0 + t_1
    if (t_2 <= (-20000.0d0)) then
        tmp = (1.0d0 / b) + ((1.0d0 / tan(b)) / ((-1.0d0) / x))
    else if (t_2 <= 2000000.0d0) then
        tmp = t_0
    else
        tmp = t_1 + (1.0d0 / b)
    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 = x * (-1.0 / Math.tan(B));
	double t_2 = t_0 + t_1;
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = (1.0 / B) + ((1.0 / Math.tan(B)) / (-1.0 / x));
	} else if (t_2 <= 2000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (1.0 / B);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x * (-1.0 / math.tan(B))
	t_2 = t_0 + t_1
	tmp = 0
	if t_2 <= -20000.0:
		tmp = (1.0 / B) + ((1.0 / math.tan(B)) / (-1.0 / x))
	elif t_2 <= 2000000.0:
		tmp = t_0
	else:
		tmp = t_1 + (1.0 / B)
	return tmp

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	t_2 = Float64(t_0 + t_1)
	tmp = 0.0
	if (t_2 <= -20000.0)
		tmp = Float64(Float64(1.0 / B) + Float64(Float64(1.0 / tan(B)) / Float64(-1.0 / x)));
	elseif (t_2 <= 2000000.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x * (-1.0 / tan(B));
	t_2 = t_0 + t_1;
	tmp = 0.0;
	if (t_2 <= -20000.0)
		tmp = (1.0 / B) + ((1.0 / tan(B)) / (-1.0 / x));
	elseif (t_2 <= 2000000.0)
		tmp = t_0;
	else
		tmp = t_1 + (1.0 / B);
	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[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -20000.0], N[(N[(1.0 / B), $MachinePrecision] + N[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2000000.0], t$95$0, N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (/.f64 #s(literal 1 binary64) (sin.f64 B))) < -2e4

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if -2e4 < (+.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 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

   if 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. Taylor expanded in B around 0

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

      1. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq -20000:\\ \;\;\;\;\frac{1}{B} + \frac{\frac{1}{\tan B}}{\frac{-1}{x}}\\ \mathbf{elif}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq 2000000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ t_2 := t\_0 + t\_1\\ t_3 := t\_1 + \frac{1}{B}\\ \mathbf{if}\;t\_2 \leq -20000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1 (* x (/ -1.0 (tan B))))
        (t_2 (+ t_0 t_1))
        (t_3 (+ t_1 (/ 1.0 B))))
   (if (<= t_2 -20000.0) t_3 (if (<= t_2 2000000.0) t_0 t_3))))

C

double code(double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x * (-1.0 / tan(B));
	double t_2 = t_0 + t_1;
	double t_3 = t_1 + (1.0 / B);
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = t_3;
	} else if (t_2 <= 2000000.0) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x * ((-1.0d0) / tan(b))
    t_2 = t_0 + t_1
    t_3 = t_1 + (1.0d0 / b)
    if (t_2 <= (-20000.0d0)) then
        tmp = t_3
    else if (t_2 <= 2000000.0d0) then
        tmp = t_0
    else
        tmp = t_3
    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 = x * (-1.0 / Math.tan(B));
	double t_2 = t_0 + t_1;
	double t_3 = t_1 + (1.0 / B);
	double tmp;
	if (t_2 <= -20000.0) {
		tmp = t_3;
	} else if (t_2 <= 2000000.0) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x * (-1.0 / math.tan(B))
	t_2 = t_0 + t_1
	t_3 = t_1 + (1.0 / B)
	tmp = 0
	if t_2 <= -20000.0:
		tmp = t_3
	elif t_2 <= 2000000.0:
		tmp = t_0
	else:
		tmp = t_3
	return tmp

Julia

function code(B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	t_2 = Float64(t_0 + t_1)
	t_3 = Float64(t_1 + Float64(1.0 / B))
	tmp = 0.0
	if (t_2 <= -20000.0)
		tmp = t_3;
	elseif (t_2 <= 2000000.0)
		tmp = t_0;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x * (-1.0 / tan(B));
	t_2 = t_0 + t_1;
	t_3 = t_1 + (1.0 / B);
	tmp = 0.0;
	if (t_2 <= -20000.0)
		tmp = t_3;
	elseif (t_2 <= 2000000.0)
		tmp = t_0;
	else
		tmp = t_3;
	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[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -20000.0], t$95$3, If[LessEqual[t$95$2, 2000000.0], t$95$0, t$95$3]]]]]]

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))) < -2e4 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. Taylor expanded in B around 0

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

      1. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -2e4 < (+.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 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq -20000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{elif}\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B} \leq 2000000:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 rewrites 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lift-tan.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-tan.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-neg-in N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. distribute-neg-frac N/A

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

   14. metadata-eval N/A

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

   15. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lift-/.f64 N/A

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

   4. neg-mul-1 N/A

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

   5. lower-fma.f64 N/A

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

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

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

   7. *-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-tan.f64 N/A

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

   12. div-inv N/A

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

   13. lift-tan.f64 N/A

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

   14. sub-neg N/A

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

   15. lift-/.f64 N/A

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

   16. lift-tan.f64 N/A

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

   17. div-inv N/A

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

   18. lift-/.f64 N/A

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

   19. *-commutative N/A

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

   20. /-rgt-identity N/A

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. sub-neg N/A

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

    4. associate-/l* N/A

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

    5. *-commutative N/A

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

    6. cancel-sign-sub-inv N/A

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

    7. lft-mult-inverse N/A

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

    8. associate-*l/ N/A

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

    9. associate-/r* N/A

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

    10. mul-1-neg N/A

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

    11. distribute-rgt-in N/A

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

    12. mul-1-neg N/A

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

    13. sub-neg N/A

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

    14. associate-/r* N/A

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

11. Applied rewrites 99.7%

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

Alternative 5: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (- (/ x (tan B)))))
   (if (<= x -1.55) t_0 (if (<= x 1.0) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = -(x / tan(B));
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / 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 = -(x / tan(b))
    if (x <= (-1.55d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = -(x / Math.tan(B));
	double tmp;
	if (x <= -1.55) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = -(x / math.tan(B))
	tmp = 0
	if x <= -1.55:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.55)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = -(x / tan(B));
	tmp = 0.0;
	if (x <= -1.55)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -1.55], t$95$0, If[LessEqual[x, 1.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 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 x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-sin.f64 97.0

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

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{x \cdot \cos B}{-\sin B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.1%

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

   if -1.55000000000000004 < x < 1

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 6e-7) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 6e-7) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 6d-7) then
        tmp = (1.0d0 - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 6e-7) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 6e-7:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 6e-7)
		tmp = Float64(Float64(1.0 - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 6e-7)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 6e-7], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.9999999999999997e-7

   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 67.3

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

   5. Applied rewrites 67.3%

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

   if 5.9999999999999997e-7 < B

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
   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 53.4

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

   5. Applied rewrites 53.4%

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

Alternative 7: 50.8% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
6. 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 51.7

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

7. Applied rewrites 51.7%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 51.8%

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

Alternative 8: 49.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -1.0) t_0 (if (<= x 1.5e-15) (/ 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 <= 1.5e-15) {
		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 <= 1.5d-15) 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 <= 1.5e-15) {
		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 <= 1.5e-15:
		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 <= 1.5e-15)
		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 <= 1.5e-15)
		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, 1.5e-15], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.5e-15 < 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 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 50.3%

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

   if -1 < x < 1.5e-15

   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 51.5

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.6%

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

Alternative 9: 50.8% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
6. 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 51.7

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

7. Applied rewrites 51.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 51.7%

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

Alternative 10: 50.8% 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 51.7

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

5. Applied rewrites 51.7%

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

Alternative 11: 25.9% 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 51.7

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

5. Applied rewrites 51.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.8%

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

Alternative 12: 3.1% accurate, 38.8× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (* B 0.16666666666666666))

C

double code(double B, double x) {
	return B * 0.16666666666666666;
}

Fortran

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

Java

public static double code(double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(B, x):
	return B * 0.16666666666666666

Julia

function code(B, x)
	return Float64(B * 0.16666666666666666)
end

MATLAB

function tmp = code(B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
6. 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 51.7

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

7. Applied rewrites 51.7%

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

8. Taylor expanded in B around inf

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

10. Applied rewrites 2.9%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 3.3%

    \[\leadsto B \cdot 0.16666666666666666 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))