Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 7.3s

Alternatives: 5

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(Float64(-fma(tan(x), 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-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-in N/A

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

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

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

   12. sqr-neg-rev N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 99.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 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-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-in N/A

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

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

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

   12. sqr-neg-rev N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 99.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-neg.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. distribute-neg-frac2 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. sqr-neg-rev N/A

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

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

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

   11. distribute-neg-in N/A

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

   12. fp-cancel-sub-sign-inv N/A

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

   13. lift-*.f64 N/A

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

   14. lift--.f64 N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 99.4

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

6. Applied rewrites 99.4%

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

Alternative 3: 55.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -0.001:\\ \;\;\;\;\frac{1 - \tan x \cdot \tan x}{1}\\ \mathbf{else}:\\ \;\;\;\;-\tanh \log \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -0.001)
   (/ (- 1.0 (* (tan x) (tan x))) 1.0)
   (- (tanh (log (tan x))))))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -0.001) {
		tmp = (1.0 - (tan(x) * tan(x))) / 1.0;
	} else {
		tmp = -tanh(log(tan(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (tan(x) <= (-0.001d0)) then
        tmp = (1.0d0 - (tan(x) * tan(x))) / 1.0d0
    else
        tmp = -tanh(log(tan(x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= -0.001) {
		tmp = (1.0 - (Math.tan(x) * Math.tan(x))) / 1.0;
	} else {
		tmp = -Math.tanh(Math.log(Math.tan(x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= -0.001:
		tmp = (1.0 - (math.tan(x) * math.tan(x))) / 1.0
	else:
		tmp = -math.tanh(math.log(math.tan(x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -0.001)
		tmp = Float64(Float64(1.0 - Float64(tan(x) * tan(x))) / 1.0);
	else
		tmp = Float64(-tanh(log(tan(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (tan(x) <= -0.001)
		tmp = (1.0 - (tan(x) * tan(x))) / 1.0;
	else
		tmp = -tanh(log(tan(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -0.001], N[(N[(1.0 - N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], (-N[Tanh[N[Log[N[Tan[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1e-3

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 20.5%

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

   if -1e-3 < (tan.f64 x)

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg2 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

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

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

      12. sqr-neg-rev N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. distribute-neg-frac2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

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

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

      11. distribute-neg-in N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. lift-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 65.7%

      \[\leadsto \color{blue}{-\tanh \log \tan x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 59.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 60.8%

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

Alternative 5: 55.3% 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
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. fp-cancel-sub-sign-inv N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-in N/A

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

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

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

   12. sqr-neg-rev N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 99.5

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-fma.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 57.0%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))