tan-example (used to crash)

Percentage Accurate: 79.8% → 99.7%

Time: 33.6s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y \cdot \tan z\\ x + \left(\frac{\tan y + \tan z}{1 - \sqrt[3]{{t_0}^{6}}} \cdot \left(1 + t_0\right) - \tan a\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (* (tan y) (tan z))))
   (+
    x
    (-
     (* (/ (+ (tan y) (tan z)) (- 1.0 (cbrt (pow t_0 6.0)))) (+ 1.0 t_0))
     (tan a)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) * tan(z);
	return x + ((((tan(y) + tan(z)) / (1.0 - cbrt(pow(t_0, 6.0)))) * (1.0 + t_0)) - tan(a));
}

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan(y) * Math.tan(z);
	return x + ((((Math.tan(y) + Math.tan(z)) / (1.0 - Math.cbrt(Math.pow(t_0, 6.0)))) * (1.0 + t_0)) - Math.tan(a));
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) * tan(z))
	return Float64(x + Float64(Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - cbrt((t_0 ^ 6.0)))) * Float64(1.0 + t_0)) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]}, N[(x + N[(N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[Power[t$95$0, 6.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. add-cube-cbrt 80.4%

      \[\leadsto x + \left(\color{blue}{\left(\sqrt[3]{\tan \left(y + z\right)} \cdot \sqrt[3]{\tan \left(y + z\right)}\right) \cdot \sqrt[3]{\tan \left(y + z\right)}} - \tan a\right) \]

   2. pow3 80.4%

      \[\leadsto x + \left(\color{blue}{{\left(\sqrt[3]{\tan \left(y + z\right)}\right)}^{3}} - \tan a\right) \]

3. Applied egg-rr 80.4%

   \[\leadsto x + \left(\color{blue}{{\left(\sqrt[3]{\tan \left(y + z\right)}\right)}^{3}} - \tan a\right) \]
4. Step-by-step derivation

   1. rem-cube-cbrt 80.7%

      \[\leadsto x + \left(\color{blue}{\tan \left(y + z\right)} - \tan a\right) \]

   2. tan-sum 99.6%

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

   3. flip-- 99.6%

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

   4. associate-/r/ 99.6%

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

   5. metadata-eval 99.6%

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

   6. pow2 99.6%

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

   7. +-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

   1. add-cbrt-cube 99.6%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\sqrt[3]{\left({\left(\tan y \cdot \tan z\right)}^{2} \cdot {\left(\tan y \cdot \tan z\right)}^{2}\right) \cdot {\left(\tan y \cdot \tan z\right)}^{2}}}} \cdot \left(\tan y \cdot \tan z + 1\right) - \tan a\right) \]

   2. pow3 99.6%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \sqrt[3]{\color{blue}{{\left({\left(\tan y \cdot \tan z\right)}^{2}\right)}^{3}}}} \cdot \left(\tan y \cdot \tan z + 1\right) - \tan a\right) \]

   3. pow-pow 99.7%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \sqrt[3]{\color{blue}{{\left(\tan y \cdot \tan z\right)}^{\left(2 \cdot 3\right)}}}} \cdot \left(\tan y \cdot \tan z + 1\right) - \tan a\right) \]

   4. metadata-eval 99.7%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \sqrt[3]{{\left(\tan y \cdot \tan z\right)}^{\color{blue}{6}}}} \cdot \left(\tan y \cdot \tan z + 1\right) - \tan a\right) \]

7. Applied egg-rr 99.7%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\sqrt[3]{{\left(\tan y \cdot \tan z\right)}^{6}}}} \cdot \left(\tan y \cdot \tan z + 1\right) - \tan a\right) \]

8. Final simplification 99.7%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \sqrt[3]{{\left(\tan y \cdot \tan z\right)}^{6}}} \cdot \left(1 + \tan y \cdot \tan z\right) - \tan a\right) \]

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (fma (+ (tan y) (tan z)) (/ 1.0 (- 1.0 (* (tan y) (tan z)))) (- (tan a)))))

C

double code(double x, double y, double z, double a) {
	return x + fma((tan(y) + tan(z)), (1.0 / (1.0 - (tan(y) * tan(z)))), -tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + fma(Float64(tan(y) + tan(z)), Float64(1.0 / Float64(1.0 - Float64(tan(y) * tan(z)))), Float64(-tan(a))))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. tan-sum 99.6%

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

   2. div-inv 99.6%

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

   3. fma-neg 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
}

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. tan-sum 56.9%

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

   2. div-inv 56.9%

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

3. Applied egg-rr 99.6%

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

   1. associate-*r/ 56.9%

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

   2. *-rgt-identity 56.9%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \]

Alternative 4: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -0.00035:\\ \;\;\;\;\left(x + t_0\right) - \tan a\\ \mathbf{elif}\;a \leq 0.0023:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= a -0.00035)
     (- (+ x t_0) (tan a))
     (if (<= a 0.0023)
       (+ (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (- x a))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (a <= -0.00035) {
		tmp = (x + t_0) - tan(a);
	} else if (a <= 0.0023) {
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + (x - a);
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (a <= (-0.00035d0)) then
        tmp = (x + t_0) - tan(a)
    else if (a <= 0.0023d0) then
        tmp = ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) + (x - a)
    else
        tmp = x + (t_0 - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (a <= -0.00035) {
		tmp = (x + t_0) - Math.tan(a);
	} else if (a <= 0.0023) {
		tmp = ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) + (x - a);
	} else {
		tmp = x + (t_0 - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if a <= -0.00035:
		tmp = (x + t_0) - math.tan(a)
	elif a <= 0.0023:
		tmp = ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) + (x - a)
	else:
		tmp = x + (t_0 - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (a <= -0.00035)
		tmp = Float64(Float64(x + t_0) - tan(a));
	elseif (a <= 0.0023)
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) + Float64(x - a));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (a <= -0.00035)
		tmp = (x + t_0) - tan(a);
	elseif (a <= 0.0023)
		tmp = ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) + (x - a);
	else
		tmp = x + (t_0 - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -0.00035], N[(N[(x + t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0023], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.49999999999999996e-4

   1. Initial program 83.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. associate-+r- 83.3%

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      2. +-commutative 83.3%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right)} - \tan a \]

   3. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right) - \tan a} \]

   if -3.49999999999999996e-4 < a < 0.0023

   1. Initial program 77.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

      3. fma-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-udef 99.8%

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

      3. div-inv 99.8%

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

      4. tan-sum 77.7%

         \[\leadsto \left(\color{blue}{\tan \left(y + z\right)} + \left(-\tan a\right)\right) + x \]

      5. sub-neg 77.7%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right)} + x \]

      6. associate--r- 77.7%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

      7. tan-sum 99.8%

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

      8. div-inv 99.8%

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

      9. fma-neg 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in a around 0 99.8%

      \[\leadsto \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \color{blue}{\left(a + -1 \cdot x\right)} \]
   9. Step-by-step derivation

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   10. Simplified 99.8%

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

   if 0.0023 < a

   1. Initial program 82.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.00035:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - \tan a\\ \mathbf{elif}\;a \leq 0.0023:\\ \;\;\;\;\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} + \left(x - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]

Alternative 5: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(y + z\right)\\ \mathbf{if}\;a \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + t_0\right) - \tan a\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (tan (+ y z))))
   (if (<= a -6.8e-7)
     (- (+ x t_0) (tan a))
     (if (<= a 1.05e-11)
       (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan((y + z));
	double tmp;
	if (a <= -6.8e-7) {
		tmp = (x + t_0) - tan(a);
	} else if (a <= 1.05e-11) {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: t_0
    real(8) :: tmp
    t_0 = tan((y + z))
    if (a <= (-6.8d-7)) then
        tmp = (x + t_0) - tan(a)
    else if (a <= 1.05d-11) then
        tmp = x + ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = x + (t_0 - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double t_0 = Math.tan((y + z));
	double tmp;
	if (a <= -6.8e-7) {
		tmp = (x + t_0) - Math.tan(a);
	} else if (a <= 1.05e-11) {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = x + (t_0 - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	t_0 = math.tan((y + z))
	tmp = 0
	if a <= -6.8e-7:
		tmp = (x + t_0) - math.tan(a)
	elif a <= 1.05e-11:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = x + (t_0 - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	t_0 = tan(Float64(y + z))
	tmp = 0.0
	if (a <= -6.8e-7)
		tmp = Float64(Float64(x + t_0) - tan(a));
	elseif (a <= 1.05e-11)
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	t_0 = tan((y + z));
	tmp = 0.0;
	if (a <= -6.8e-7)
		tmp = (x + t_0) - tan(a);
	elseif (a <= 1.05e-11)
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	else
		tmp = x + (t_0 - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, -6.8e-7], N[(N[(x + t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.05e-11], N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.79999999999999948e-7

   1. Initial program 83.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. associate-+r- 83.3%

         \[\leadsto \color{blue}{\left(x + \tan \left(y + z\right)\right) - \tan a} \]

      2. +-commutative 83.3%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right)} - \tan a \]

   3. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\left(\tan \left(y + z\right) + x\right) - \tan a} \]

   if -6.79999999999999948e-7 < a < 1.0499999999999999e-11

   1. Initial program 77.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 77.9%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 77.9%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in a around 0 77.5%

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

      1. neg-mul-1 77.5%

         \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]

   6. Simplified 77.5%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]
   7. Step-by-step derivation

      1. sub-neg 77.5%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]

      2. add-sqr-sqrt 0.0%

         \[\leadsto \tan \left(y + z\right) + \left(-\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) \]

      3. sqrt-unprod 2.7%

         \[\leadsto \tan \left(y + z\right) + \left(-\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) \]

      4. sqr-neg 2.7%

         \[\leadsto \tan \left(y + z\right) + \left(-\sqrt{\color{blue}{x \cdot x}}\right) \]

      5. sqrt-unprod 2.7%

         \[\leadsto \tan \left(y + z\right) + \left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \]

      6. add-sqr-sqrt 2.7%

         \[\leadsto \tan \left(y + z\right) + \left(-\color{blue}{x}\right) \]

      7. add-sqr-sqrt 0.0%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}} \]

      8. sqrt-unprod 77.5%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]

      9. sqr-neg 77.5%

         \[\leadsto \tan \left(y + z\right) + \sqrt{\color{blue}{x \cdot x}} \]

      10. sqrt-unprod 77.0%

          \[\leadsto \tan \left(y + z\right) + \color{blue}{\sqrt{x} \cdot \sqrt{x}} \]

      11. add-sqr-sqrt 77.5%

          \[\leadsto \tan \left(y + z\right) + \color{blue}{x} \]

   8. Applied egg-rr 77.5%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + x} \]
   9. Step-by-step derivation

      1. tan-sum 98.6%

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

      2. div-inv 98.6%

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

   10. Applied egg-rr 98.6%

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

       1. associate-*r/ 98.6%

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

       2. *-rgt-identity 98.6%

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

   12. Simplified 98.6%

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

   if 1.0499999999999999e-11 < a

   1. Initial program 82.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \tan \left(y + z\right)\right) - \tan a\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \end{array} \]

Alternative 6: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Final simplification 80.7%

   \[\leadsto x + \left(\tan \left(y + z\right) - \tan a\right) \]

Alternative 7: 51.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + \tan \left(y + z\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (tan (+ y z))))

C

double code(double x, double y, double z, double a) {
	return x + tan((y + z));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + tan((y + z))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + Math.tan((y + z));
}

Python

def code(x, y, z, a):
	return x + math.tan((y + z))

Julia

function code(x, y, z, a)
	return Float64(x + tan(Float64(y + z)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + tan((y + z));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. +-commutative 80.7%

      \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

   2. associate-+l- 80.6%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

3. Applied egg-rr 80.6%

   \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

4. Taylor expanded in a around 0 47.1%

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

   1. neg-mul-1 47.1%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]

6. Simplified 47.1%

   \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]
7. Step-by-step derivation

   1. sub-neg 47.1%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]

   2. add-sqr-sqrt 0.0%

      \[\leadsto \tan \left(y + z\right) + \left(-\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right) \]

   3. sqrt-unprod 3.1%

      \[\leadsto \tan \left(y + z\right) + \left(-\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) \]

   4. sqr-neg 3.1%

      \[\leadsto \tan \left(y + z\right) + \left(-\sqrt{\color{blue}{x \cdot x}}\right) \]

   5. sqrt-unprod 3.1%

      \[\leadsto \tan \left(y + z\right) + \left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \]

   6. add-sqr-sqrt 3.1%

      \[\leadsto \tan \left(y + z\right) + \left(-\color{blue}{x}\right) \]

   7. add-sqr-sqrt 0.0%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}} \]

   8. sqrt-unprod 47.1%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]

   9. sqr-neg 47.1%

      \[\leadsto \tan \left(y + z\right) + \sqrt{\color{blue}{x \cdot x}} \]

   10. sqrt-unprod 46.9%

       \[\leadsto \tan \left(y + z\right) + \color{blue}{\sqrt{x} \cdot \sqrt{x}} \]

   11. add-sqr-sqrt 47.1%

       \[\leadsto \tan \left(y + z\right) + \color{blue}{x} \]

8. Applied egg-rr 47.1%

   \[\leadsto \color{blue}{\tan \left(y + z\right) + x} \]

9. Final simplification 47.1%

   \[\leadsto x + \tan \left(y + z\right) \]

Alternative 8: 31.9% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 x)

C

double code(double x, double y, double z, double a) {
	return x;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x
end function

Java

public static double code(double x, double y, double z, double a) {
	return x;
}

Python

def code(x, y, z, a):
	return x

Julia

function code(x, y, z, a)
	return x
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 80.7%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Taylor expanded in x around inf 29.5%

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

3. Final simplification 29.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023312 
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))