Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.1s

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 99.5

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

4. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
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. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 2: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;t\_0 \leq 0.6:\\ \;\;\;\;\frac{1 + \left(0.5 \cdot \cos \left(x + x\right) - 0.5\right)}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 - {\tan x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= t_0 0.6)
     (/ (+ 1.0 (- (* 0.5 (cos (+ x x))) 0.5)) (+ 1.0 t_0))
     (- 1.0 (pow (tan x) 2.0)))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	double tmp;
	if (t_0 <= 0.6) {
		tmp = (1.0 + ((0.5 * cos((x + x))) - 0.5)) / (1.0 + t_0);
	} else {
		tmp = 1.0 - pow(tan(x), 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) * tan(x)
    if (t_0 <= 0.6d0) then
        tmp = (1.0d0 + ((0.5d0 * cos((x + x))) - 0.5d0)) / (1.0d0 + t_0)
    else
        tmp = 1.0d0 - (tan(x) ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	double tmp;
	if (t_0 <= 0.6) {
		tmp = (1.0 + ((0.5 * Math.cos((x + x))) - 0.5)) / (1.0 + t_0);
	} else {
		tmp = 1.0 - Math.pow(Math.tan(x), 2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	tmp = 0
	if t_0 <= 0.6:
		tmp = (1.0 + ((0.5 * math.cos((x + x))) - 0.5)) / (1.0 + t_0)
	else:
		tmp = 1.0 - math.pow(math.tan(x), 2.0)
	return tmp

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	tmp = 0.0
	if (t_0 <= 0.6)
		tmp = Float64(Float64(1.0 + Float64(Float64(0.5 * cos(Float64(x + x))) - 0.5)) / Float64(1.0 + t_0));
	else
		tmp = Float64(1.0 - (tan(x) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = tan(x) * tan(x);
	tmp = 0.0;
	if (t_0 <= 0.6)
		tmp = (1.0 + ((0.5 * cos((x + x))) - 0.5)) / (1.0 + t_0);
	else
		tmp = 1.0 - (tan(x) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.6], N[(N[(1.0 + N[(N[(0.5 * N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 0.599999999999999978

   1. Initial program 99.7%

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

      1. tan-quot N/A

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

      2. div-inv N/A

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

      3. tan-quot N/A

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

      4. div-inv N/A

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

      5. swap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. cos-2 N/A

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

      10. cos-sum N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. inv-pow N/A

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

      15. inv-pow N/A

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

      16. pow-prod-down N/A

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

      17. inv-pow N/A

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

      18. lower-/.f64 N/A

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

      19. sqr-cos-a N/A

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

      20. lower-+.f64 N/A

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

      21. cos-2 N/A

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

      22. cos-sum N/A

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

      23. lower-*.f64 N/A

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

   4. Applied egg-rr 99.7%

      \[\leadsto \frac{1 - \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \frac{1}{0.5 + 0.5 \cdot \cos \left(x + x\right)}}}{1 + \tan x \cdot \tan x} \]

   5. Taylor expanded in x around 0

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

   7. Simplified 78.9%

      \[\leadsto \frac{1 - \left(0.5 - 0.5 \cdot \cos \left(x + x\right)\right) \cdot \color{blue}{1}}{1 + \tan x \cdot \tan x} \]

   if 0.599999999999999978 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 98.9%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   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. pow2 N/A

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

      4. lift-pow.f64 98.9

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 16.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;\frac{1 + \left(0.5 \cdot \cos \left(x + x\right) - 0.5\right)}{1 + \tan x \cdot \tan x}\\ \mathbf{else}:\\ \;\;\;\;1 - {\tan x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;{\left(1 + t\_0\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0)))
   (if (<= (* (tan x) (tan x)) 0.6) (pow (+ 1.0 t_0) -2.0) (- 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	double tmp;
	if ((tan(x) * tan(x)) <= 0.6) {
		tmp = pow((1.0 + t_0), -2.0);
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan(x) ** 2.0d0
    if ((tan(x) * tan(x)) <= 0.6d0) then
        tmp = (1.0d0 + t_0) ** (-2.0d0)
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	double tmp;
	if ((Math.tan(x) * Math.tan(x)) <= 0.6) {
		tmp = Math.pow((1.0 + t_0), -2.0);
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	tmp = 0
	if (math.tan(x) * math.tan(x)) <= 0.6:
		tmp = math.pow((1.0 + t_0), -2.0)
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	tmp = 0.0
	if (Float64(tan(x) * tan(x)) <= 0.6)
		tmp = Float64(1.0 + t_0) ^ -2.0;
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = 0.0;
	if ((tan(x) * tan(x)) <= 0.6)
		tmp = (1.0 + t_0) ^ -2.0;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision], 0.6], N[Power[N[(1.0 + t$95$0), $MachinePrecision], -2.0], $MachinePrecision], N[(1.0 - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 x) (tan.f64 x)) < 0.599999999999999978

   1. Initial program 99.7%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   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 99.7

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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(1 - {\tan x}^{4}\right) \cdot {\left({\tan x}^{2} + 1\right)}^{-2}} \]

   7. Taylor expanded in x around 0

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

   9. Simplified 78.9%

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

   if 0.599999999999999978 < (*.f64 (tan.f64 x) (tan.f64 x))

   1. Initial program 98.9%

      \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
   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 98.9

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

   4. Applied egg-rr 98.9%

      \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
   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. pow2 N/A

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

      4. lift-pow.f64 98.9

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 16.8%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \cdot \tan x \leq 0.6:\\ \;\;\;\;{\left(1 + {\tan x}^{2}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;1 - {\tan x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - {\tan x}^{2}}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (pow (tan x) 2.0)) (fma (tan x) (tan x) 1.0)))

C

double code(double x) {
	return (1.0 - pow(tan(x), 2.0)) / fma(tan(x), tan(x), 1.0);
}

Julia

function code(x)
	return Float64(Float64(1.0 - (tan(x) ^ 2.0)) / fma(tan(x), tan(x), 1.0))
end

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $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 99.5

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

4. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
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. pow2 N/A

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

   4. lift-pow.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t\_0}{1 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	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) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$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 99.5

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

4. Applied egg-rr 99.5%

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

   1. lift-tan.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. pow-pow N/A

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

   6. inv-pow N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift-/.f64 99.3

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

   \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}} \]
8. Add Preprocessing

Alternative 6: 59.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 1 - {\tan x}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (pow (tan x) 2.0)))

C

double code(double x) {
	return 1.0 - pow(tan(x), 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (tan(x) ** 2.0d0)
end function

Java

public static double code(double x) {
	return 1.0 - Math.pow(Math.tan(x), 2.0);
}

Python

def code(x):
	return 1.0 - math.pow(math.tan(x), 2.0)

Julia

function code(x)
	return Float64(1.0 - (tan(x) ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (tan(x) ^ 2.0);
end

Wolfram

code[x_] := N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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 99.5

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

4. Applied egg-rr 99.5%

   \[\leadsto \frac{1 - \tan x \cdot \tan x}{\color{blue}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}} \]
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. pow2 N/A

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

   4. lift-pow.f64 99.5

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0

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

9. Simplified 57.8%

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

10. Final simplification 57.8%

    \[\leadsto 1 - {\tan x}^{2} \]
11. Add Preprocessing

Alternative 7: 58.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 1 - {\tan x}^{4} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (pow (tan x) 4.0)))

C

double code(double x) {
	return 1.0 - pow(tan(x), 4.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (tan(x) ** 4.0d0)
end function

Java

public static double code(double x) {
	return 1.0 - Math.pow(Math.tan(x), 4.0);
}

Python

def code(x):
	return 1.0 - math.pow(math.tan(x), 4.0)

Julia

function code(x)
	return Float64(1.0 - (tan(x) ^ 4.0))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (tan(x) ^ 4.0);
end

Wolfram

code[x_] := N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 4.0], $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 99.5

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

4. Applied egg-rr 99.5%

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

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(1 - {\tan x}^{4}\right) \cdot {\left({\tan x}^{2} + 1\right)}^{-2}} \]

7. Taylor expanded in x around 0

   \[\leadsto \left(1 - {\tan x}^{4}\right) \cdot \color{blue}{1} \]
8. Step-by-step derivation

9. Simplified 57.0%

   \[\leadsto \left(1 - {\tan x}^{4}\right) \cdot \color{blue}{1} \]

10. Final simplification 57.0%

    \[\leadsto 1 - {\tan x}^{4} \]
11. Add Preprocessing

Alternative 8: 55.1% accurate, 428.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.4%

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

4. Applied egg-rr 53.4%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))