Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 13.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (tan x) (- (tan x)) 1.0) (fma (tan x) (tan x) 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 99.4

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

4. Applied egg-rr 99.4%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

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

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

   4. lift-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lift-fma.f64 99.4

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

6. Applied egg-rr 99.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]
7. Add Preprocessing

Alternative 2: 56.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq -0.02:\\ \;\;\;\;\frac{1 - x \cdot x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) -0.02)
     (/ (- 1.0 (* x x)) (fma x x 1.0))
     1.0)))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (((1.0 - t_0) / (1.0 + t_0)) <= -0.02) {
		tmp = (1.0 - (x * x)) / fma(x, x, 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0)) <= -0.02)
		tmp = Float64(Float64(1.0 - Float64(x * x)) / fma(x, x, 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x)))) < -0.0200000000000000004

   1. Initial program 99.2%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-tan.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.2

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

   4. Applied egg-rr 99.2%

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

      1. lift-tan.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 99.2

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 99.2

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{1 + -1 \cdot {x}^{2}}}{{\tan x}^{2} + 1} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{1 - {x}^{2}}}{{\tan x}^{2} + 1} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{1 - {x}^{2}}}{{\tan x}^{2} + 1} \]

      4. unpow2 N/A

         \[\leadsto \frac{1 - \color{blue}{x \cdot x}}{{\tan x}^{2} + 1} \]

      5. lower-*.f64 4.3

         \[\leadsto \frac{1 - \color{blue}{x \cdot x}}{{\tan x}^{2} + 1} \]

   9. Simplified 4.3%

      \[\leadsto \frac{\color{blue}{1 - x \cdot x}}{{\tan x}^{2} + 1} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{1 - x \cdot x}{\color{blue}{1 + {x}^{2}}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{1 - x \cdot x}{\color{blue}{{x}^{2} + 1}} \]

       2. unpow2 N/A

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

       3. lower-fma.f64 10.6

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

   12. Simplified 10.6%

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

   if -0.0200000000000000004 < (/.f64 (-.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))) (+.f64 #s(literal 1 binary64) (*.f64 (tan.f64 x) (tan.f64 x))))

   1. Initial program 99.6%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   2. Add Preprocessing

   4. Applied egg-rr 72.5%

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

Reproduce

herbie shell --seed 2024218 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))