Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 3.2s

Alternatives: 8

Speedup: 1.4×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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, 1.2× 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 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (t_0 - -1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (1.0d0 - t_0) / (t_0 - (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (t_0 - -1.0);
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[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-fma.f64 N/A

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

   2. pow2 N/A

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

   3. lift-pow.f64 N/A

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

   4. add-flip N/A

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

   5. metadata-eval N/A

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

   6. lift--.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 3: 56.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-\tanh \log \left(\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x\right)\\ \mathbf{elif}\;\tan x \leq -0.02:\\ \;\;\;\;\frac{--1}{{\tan x}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;-\tanh \log \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -1.0)
   (- (tanh (log (* (fma (* x x) 0.3333333333333333 1.0) x))))
   (if (<= (tan x) -0.02)
     (/ (- -1.0) (- (pow (tan x) 2.0) -1.0))
     (- (tanh (log (tan x)))))))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -1.0) {
		tmp = -tanh(log((fma((x * x), 0.3333333333333333, 1.0) * x)));
	} else if (tan(x) <= -0.02) {
		tmp = -(-1.0) / (pow(tan(x), 2.0) - -1.0);
	} else {
		tmp = -tanh(log(tan(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -1.0)
		tmp = Float64(-tanh(log(Float64(fma(Float64(x * x), 0.3333333333333333, 1.0) * x))));
	elseif (tan(x) <= -0.02)
		tmp = Float64(Float64(-(-1.0)) / Float64((tan(x) ^ 2.0) - -1.0));
	else
		tmp = Float64(-tanh(log(tan(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -1.0], (-N[Tanh[N[Log[N[(N[(N[(x * x), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Tan[x], $MachinePrecision], -0.02], N[((--1.0) / N[(N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], (-N[Tanh[N[Log[N[Tan[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{1 + \tan x \cdot \tan x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{1 + \tan x \cdot \tan x} \]

      13. lower-log.f64 48.8

          \[\leadsto \frac{-\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{1 + \tan x \cdot \tan x} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{1 + \tan x \cdot \tan x}} \]

      15. +-commutative N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\tan x \cdot \tan x - \color{blue}{-1}} \]

      18. lower--.f64 48.8

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - -1}} \]

   3. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{{\tan x}^{2} - -1}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)} \cdot 2\right)}{{\tan x}^{2} - -1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {x}^{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{x}^{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      4. lower-pow.f64 24.6

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

   6. Applied rewrites 24.6%

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \color{blue}{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)} \cdot 2\right)}{{\tan x}^{2} - -1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}}^{2} - -1} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right)}^{2} - -1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {x}^{2}}\right)\right)}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{x}^{2}}\right)\right)}^{2} - -1} \]

      4. lower-pow.f64 24.5

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right)}^{2} - -1} \]

   9. Applied rewrites 24.5%

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}}^{2} - -1} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)\right)}}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1} \]

       3. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}\right)} \]

       4. lower-neg.f64 N/A

          \[\leadsto \color{blue}{-\frac{\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}} \]

   11. Applied rewrites 26.9%

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

   if -1 < (tan.f64 x) < -0.0200000000000000004

   1. Initial program 99.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{1 + \tan x \cdot \tan x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{1 + \tan x \cdot \tan x} \]

      13. lower-log.f64 48.8

          \[\leadsto \frac{-\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{1 + \tan x \cdot \tan x} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{1 + \tan x \cdot \tan x}} \]

      15. +-commutative N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\tan x \cdot \tan x - \color{blue}{-1}} \]

      18. lower--.f64 48.8

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - -1}} \]

   3. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{{\tan x}^{2} - -1}} \]

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 54.7%

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

   if -0.0200000000000000004 < (tan.f64 x)

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.5

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-expm1.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]

      10. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right)} \]

      11. lift-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x + 1}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x} + 1}\right) \]

      13. add-flip N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\tan x \cdot \tan x - \color{blue}{-1}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x} - -1}\right) \]

      16. pow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{{\tan x}^{2}} - -1}\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{{\tan x}^{2}} - -1}\right) \]

   5. Applied rewrites 48.9%

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

Alternative 4: 56.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-\tanh \log \left(\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x\right)\\ \mathbf{elif}\;\tan x \leq -0.02:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-\tanh \log \tan x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -1.0)
   (- (tanh (log (* (fma (* x x) 0.3333333333333333 1.0) x))))
   (if (<= (tan x) -0.02) 1.0 (- (tanh (log (tan x)))))))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -1.0) {
		tmp = -tanh(log((fma((x * x), 0.3333333333333333, 1.0) * x)));
	} else if (tan(x) <= -0.02) {
		tmp = 1.0;
	} else {
		tmp = -tanh(log(tan(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -1.0)
		tmp = Float64(-tanh(log(Float64(fma(Float64(x * x), 0.3333333333333333, 1.0) * x))));
	elseif (tan(x) <= -0.02)
		tmp = 1.0;
	else
		tmp = Float64(-tanh(log(tan(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -1.0], (-N[Tanh[N[Log[N[(N[(N[(x * x), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[Tan[x], $MachinePrecision], -0.02], 1.0, (-N[Tanh[N[Log[N[Tan[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{1 + \tan x \cdot \tan x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{1 + \tan x \cdot \tan x} \]

      13. lower-log.f64 48.8

          \[\leadsto \frac{-\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{1 + \tan x \cdot \tan x} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{1 + \tan x \cdot \tan x}} \]

      15. +-commutative N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\tan x \cdot \tan x - \color{blue}{-1}} \]

      18. lower--.f64 48.8

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - -1}} \]

   3. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{{\tan x}^{2} - -1}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)} \cdot 2\right)}{{\tan x}^{2} - -1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {x}^{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{x}^{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      4. lower-pow.f64 24.6

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

   6. Applied rewrites 24.6%

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \color{blue}{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)} \cdot 2\right)}{{\tan x}^{2} - -1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}}^{2} - -1} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right)}^{2} - -1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {x}^{2}}\right)\right)}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{x}^{2}}\right)\right)}^{2} - -1} \]

      4. lower-pow.f64 24.5

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right)}^{2} - -1} \]

   9. Applied rewrites 24.5%

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}}^{2} - -1} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)\right)}}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1} \]

       3. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}\right)} \]

       4. lower-neg.f64 N/A

          \[\leadsto \color{blue}{-\frac{\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}} \]

   11. Applied rewrites 26.9%

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

   if -1 < (tan.f64 x) < -0.0200000000000000004

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.3%

      \[\leadsto \color{blue}{1} \]

   if -0.0200000000000000004 < (tan.f64 x)

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.5

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. lift-log.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-expm1.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)} \]

      10. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\mathsf{fma}\left(\tan x, \tan x, 1\right)}\right)} \]

      11. lift-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x + 1}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x} + 1}\right) \]

      13. add-flip N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\tan x \cdot \tan x - \color{blue}{-1}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x} - -1}\right) \]

      16. pow2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{{\tan x}^{2}} - -1}\right) \]

      17. lift-pow.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{{\tan x}^{2}} - -1}\right) \]

   5. Applied rewrites 48.9%

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

Alternative 5: 56.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq -0.04:\\ \;\;\;\;-\tanh \log \left(\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) -0.04)
     (- (tanh (log (* (fma (* x x) 0.3333333333333333 1.0) x))))
     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)) <= -0.04) {
		tmp = -tanh(log((fma((x * x), 0.3333333333333333, 1.0) * x)));
	} 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)) <= -0.04)
		tmp = Float64(-tanh(log(Float64(fma(Float64(x * x), 0.3333333333333333, 1.0) * x))));
	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], -0.04], (-N[Tanh[N[Log[N[(N[(N[(x * x), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision]], $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)))) < -0.0400000000000000008

   1. Initial program 99.5%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. sub-negate-rev N/A

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

      4. lift--.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-negate-rev N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. pow-to-exp N/A

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

      11. lower-expm1.f64 N/A

          \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\log \tan x \cdot 2\right)}}{1 + \tan x \cdot \tan x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\color{blue}{\log \tan x \cdot 2}\right)}{1 + \tan x \cdot \tan x} \]

      13. lower-log.f64 48.8

          \[\leadsto \frac{-\mathsf{expm1}\left(\color{blue}{\log \tan x} \cdot 2\right)}{1 + \tan x \cdot \tan x} \]

      14. lift-+.f64 N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{1 + \tan x \cdot \tan x}} \]

      15. +-commutative N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x + 1}} \]

      16. add-flip N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - \left(\mathsf{neg}\left(1\right)\right)}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\tan x \cdot \tan x - \color{blue}{-1}} \]

      18. lower--.f64 48.8

          \[\leadsto \frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{\color{blue}{\tan x \cdot \tan x - -1}} \]

   3. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\frac{-\mathsf{expm1}\left(\log \tan x \cdot 2\right)}{{\tan x}^{2} - -1}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)} \cdot 2\right)}{{\tan x}^{2} - -1} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {x}^{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{x}^{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

      4. lower-pow.f64 24.6

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right) \cdot 2\right)}{{\tan x}^{2} - -1} \]

   6. Applied rewrites 24.6%

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \color{blue}{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)} \cdot 2\right)}{{\tan x}^{2} - -1} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}}^{2} - -1} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {x}^{2}\right)}\right)}^{2} - -1} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {x}^{2}}\right)\right)}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{x}^{2}}\right)\right)}^{2} - -1} \]

      4. lower-pow.f64 24.5

         \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{\color{blue}{2}}\right)\right)}^{2} - -1} \]

   9. Applied rewrites 24.5%

      \[\leadsto \frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\color{blue}{\left(x \cdot \left(1 + 0.3333333333333333 \cdot {x}^{2}\right)\right)}}^{2} - -1} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}} \]

       2. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)\right)}}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1} \]

       3. distribute-frac-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}\right)} \]

       4. lower-neg.f64 N/A

          \[\leadsto \color{blue}{-\frac{\mathsf{expm1}\left(\log \left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right) \cdot 2\right)}{{\left(x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)\right)}^{2} - -1}} \]

   11. Applied rewrites 26.9%

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

   if -0.0400000000000000008 < (/.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 99.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.3%

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

Alternative 6: 54.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq -0.04:\\ \;\;\;\;\frac{1 - x}{\mathsf{fma}\left(x, x, 1\right)} \cdot \left(x - -1\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) -0.04)
     (* (/ (- 1.0 x) (fma x x 1.0)) (- 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)) <= -0.04) {
		tmp = ((1.0 - x) / fma(x, x, 1.0)) * (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)) <= -0.04)
		tmp = Float64(Float64(Float64(1.0 - x) / fma(x, x, 1.0)) * Float64(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], -0.04], N[(N[(N[(1.0 - x), $MachinePrecision] / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(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)))) < -0.0400000000000000008

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 50.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 50.4%

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

   13. Taylor expanded in x around 0

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

   15. Applied rewrites 51.6%

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

       1. lift--.f64 N/A

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

       2. metadata-eval N/A

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

       3. lift-*.f64 N/A

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

       4. difference-of-squares N/A

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

       5. lower-*.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower--.f64 51.6

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

   17. Applied rewrites 51.6%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 52.4

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

       7. lift-+.f64 N/A

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

       8. +-commutative N/A

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

       9. add-flip N/A

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

       10. metadata-eval N/A

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

       11. lower--.f64 52.4

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

   19. Applied rewrites 52.4%

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

   if -0.0400000000000000008 < (/.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 99.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.3%

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

Alternative 7: 53.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \mathbf{if}\;\frac{1 - t\_0}{1 + t\_0} \leq -0.04:\\ \;\;\;\;\frac{1 - x \cdot x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x))))
   (if (<= (/ (- 1.0 t_0) (+ 1.0 t_0)) -0.04)
     (/ (- 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)) <= -0.04) {
		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)) <= -0.04)
		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], -0.04], 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)))) < -0.0400000000000000008

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 50.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 50.4%

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

   13. Taylor expanded in x around 0

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

   15. Applied rewrites 51.6%

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

   if -0.0400000000000000008 < (/.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 99.5%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.3%

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

Alternative 8: 53.0% accurate, 155.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 99.5%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 54.3%

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

Reproduce

herbie shell --seed 2025140 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))