tan-example (used to crash)

Percentage Accurate: 80.2% → 99.7%

Time: 30.6s

Alternatives: 14

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: 80.2% 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, 0.4× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (- (* (+ (tan y) (tan z)) (/ -1.0 (+ -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 / (-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) / ((-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 / (-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 / (-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(-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 / (-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[(-1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. 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. fmm-def 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. Applied egg-rr 99.8%

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

   1. fmm-undef 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ t_1 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, t\_0\right)\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x - \left(a + t\_1 \cdot \frac{1}{-1 + \tan y \cdot \tan z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))) (t_1 (+ (tan y) (tan z))))
   (if (<= (tan a) -0.02)
     (fma t_1 1.0 t_0)
     (if (<= (tan a) 1e-12)
       (- x (+ a (* t_1 (/ 1.0 (+ -1.0 (* (tan y) (tan z)))))))
       (fma (sin (+ y z)) (/ 1.0 (cos (+ y z))) t_0)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double t_1 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = fma(t_1, 1.0, t_0);
	} else if (tan(a) <= 1e-12) {
		tmp = x - (a + (t_1 * (1.0 / (-1.0 + (tan(y) * tan(z))))));
	} else {
		tmp = fma(sin((y + z)), (1.0 / cos((y + z))), t_0);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	t_1 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = fma(t_1, 1.0, t_0);
	elseif (tan(a) <= 1e-12)
		tmp = Float64(x - Float64(a + Float64(t_1 * Float64(1.0 / Float64(-1.0 + Float64(tan(y) * tan(z)))))));
	else
		tmp = fma(sin(Float64(y + z)), Float64(1.0 / cos(Float64(y + z))), t_0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-12], N[(x - N[(a + N[(t$95$1 * N[(1.0 / N[(-1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0200000000000000004

   1. Initial program 79.0%

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

      1. +-commutative 79.0%

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

      2. associate-+l- 78.7%

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

      3. tan-sum 99.5%

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

      4. div-inv 99.5%

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

      5. fmm-def 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0 79.5%

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

   if -0.0200000000000000004 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 74.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 74.0%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.8%

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

      3. fmm-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. fmm-undef 99.8%

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

   7. Simplified 99.8%

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

   if 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 70.8%

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

      1. +-commutative 70.8%

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

      2. associate-+l- 70.8%

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

      3. tan-quot 70.8%

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

      4. div-inv 70.8%

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

      5. fmm-def 70.9%

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

   4. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, -\left(\tan a - x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, 1, x - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x - \left(a + \left(\tan y + \tan z\right) \cdot \frac{1}{-1 + \tan y \cdot \tan z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \tan a\\ t_1 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, t\_0\right)\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x - \left(a + \frac{t\_1}{-1 + \tan y \cdot \tan z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (- x (tan a))) (t_1 (+ (tan y) (tan z))))
   (if (<= (tan a) -0.02)
     (fma t_1 1.0 t_0)
     (if (<= (tan a) 1e-12)
       (- x (+ a (/ t_1 (+ -1.0 (* (tan y) (tan z))))))
       (fma (sin (+ y z)) (/ 1.0 (cos (+ y z))) t_0)))))

C

double code(double x, double y, double z, double a) {
	double t_0 = x - tan(a);
	double t_1 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -0.02) {
		tmp = fma(t_1, 1.0, t_0);
	} else if (tan(a) <= 1e-12) {
		tmp = x - (a + (t_1 / (-1.0 + (tan(y) * tan(z)))));
	} else {
		tmp = fma(sin((y + z)), (1.0 / cos((y + z))), t_0);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(x - tan(a))
	t_1 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -0.02)
		tmp = fma(t_1, 1.0, t_0);
	elseif (tan(a) <= 1e-12)
		tmp = Float64(x - Float64(a + Float64(t_1 / Float64(-1.0 + Float64(tan(y) * tan(z))))));
	else
		tmp = fma(sin(Float64(y + z)), Float64(1.0 / cos(Float64(y + z))), t_0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 1e-12], N[(x - N[(a + N[(t$95$1 / N[(-1.0 + N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0200000000000000004

   1. Initial program 79.0%

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

      1. +-commutative 79.0%

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

      2. associate-+l- 78.7%

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

      3. tan-sum 99.5%

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

      4. div-inv 99.5%

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

      5. fmm-def 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in y around 0 79.5%

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

   if -0.0200000000000000004 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 74.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 74.0%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.8%

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

      3. fmm-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. fmm-undef 99.8%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   if 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 70.8%

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

      1. +-commutative 70.8%

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

      2. associate-+l- 70.8%

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

      3. tan-quot 70.8%

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

      4. div-inv 70.8%

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

      5. fmm-def 70.9%

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

   4. Applied egg-rr 70.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, -\left(\tan a - x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\tan y + \tan z, 1, x - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 10^{-12}:\\ \;\;\;\;x - \left(a + \frac{\tan y + \tan z}{-1 + \tan y \cdot \tan z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, x - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return x - (tan(a) + ((tan(y) + tan(z)) / (-1.0 + (tan(y) * tan(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(a) + ((tan(y) + tan(z)) / ((-1.0d0) + (tan(y) * tan(z)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. +-commutative 74.5%

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

   2. sub-neg 74.5%

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

   3. associate-+l+ 74.4%

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

   4. tan-sum 99.7%

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

   5. div-inv 99.7%

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

   6. fma-define 99.7%

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

   7. neg-mul-1 99.7%

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

   8. fma-define 99.7%

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

4. Applied egg-rr 99.7%

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

   1. fma-undefine 99.7%

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

   2. fma-undefine 99.7%

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

   3. neg-mul-1 99.7%

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

   4. associate-+r+ 99.8%

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

   5. sub-neg 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 5: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-5} \lor \neg \left(\tan a \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -5e-5) (not (<= (tan a) 5e-54)))
   (+ x (- (tan y) (tan a)))
   (+ x (- (tan (+ y z)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -5e-5) || !(tan(a) <= 5e-54)) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan((y + z)) - 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) :: tmp
    if ((tan(a) <= (-5d-5)) .or. (.not. (tan(a) <= 5d-54))) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan((y + z)) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((Math.tan(a) <= -5e-5) || !(Math.tan(a) <= 5e-54)) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan((y + z)) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -5e-5) or not (math.tan(a) <= 5e-54):
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan((y + z)) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -5e-5) || !(tan(a) <= 5e-54))
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -5e-5) || ~((tan(a) <= 5e-54)))
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan((y + z)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -5e-5], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 5e-54]], $MachinePrecision]], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -5.00000000000000024e-5 or 5.00000000000000015e-54 < (tan.f64 a)

   1. Initial program 74.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 57.2%

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

   if -5.00000000000000024e-5 < (tan.f64 a) < 5.00000000000000015e-54

   1. Initial program 75.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 75.0%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -5 \cdot 10^{-5} \lor \neg \left(\tan a \leq 5 \cdot 10^{-54}\right):\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. +-commutative 74.5%

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

   2. associate-+l- 74.4%

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

   3. tan-sum 99.7%

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

   4. div-inv 99.7%

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

   5. fmm-def 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in y around 0 75.1%

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

6. Final simplification 75.1%

   \[\leadsto \mathsf{fma}\left(\tan y + \tan z, 1, x - \tan a\right) \]
7. Add Preprocessing

Alternative 7: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -5 \cdot 10^{-13}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -5e-13) (+ x (- (tan y) (tan a))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-13) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (tan(z) - 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) :: tmp
    if ((y + z) <= (-5d-13)) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -5e-13) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -5e-13:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -5e-13)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -5e-13)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -5e-13], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -4.9999999999999999e-13

   1. Initial program 70.2%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 46.7%

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

   if -4.9999999999999999e-13 < (+.f64 y z)

   1. Initial program 77.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 63.8%

      \[\leadsto x + \left(\tan \color{blue}{z} - \tan a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{3}\\ \mathbf{elif}\;a \leq 195000000:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.9)
   (pow (cbrt x) 3.0)
   (if (<= a 195000000.0) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.9) {
		tmp = pow(cbrt(x), 3.0);
	} else if (a <= 195000000.0) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.9) {
		tmp = Math.pow(Math.cbrt(x), 3.0);
	} else if (a <= 195000000.0) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.9)
		tmp = cbrt(x) ^ 3.0;
	elseif (a <= 195000000.0)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.9], N[Power[N[Power[x, 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[a, 195000000.0], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8999999999999999

   1. Initial program 76.5%

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

      1. add-cube-cbrt 75.4%

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

      2. pow3 75.4%

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

      3. +-commutative 75.4%

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

      4. associate-+l- 75.3%

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

   4. Applied egg-rr 75.3%

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

   5. Taylor expanded in x around inf 23.5%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{x}\right)}}^{3} \]

   if -1.8999999999999999 < a < 1.95e8

   1. Initial program 74.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 73.3%

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

   if 1.95e8 < a

   1. Initial program 71.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 23.1%

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

Alternative 9: 80.2% 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 74.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 10: 41.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-27}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-214}:\\ \;\;\;\;x + \left(\tan z - a\right)\\ \mathbf{elif}\;a \leq 195000000:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -9.5e-27)
   x
   (if (<= a -1.55e-214)
     (+ x (- (tan z) a))
     (if (<= a 195000000.0) (+ x (- (tan y) a)) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -9.5e-27) {
		tmp = x;
	} else if (a <= -1.55e-214) {
		tmp = x + (tan(z) - a);
	} else if (a <= 195000000.0) {
		tmp = x + (tan(y) - a);
	} else {
		tmp = x;
	}
	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) :: tmp
    if (a <= (-9.5d-27)) then
        tmp = x
    else if (a <= (-1.55d-214)) then
        tmp = x + (tan(z) - a)
    else if (a <= 195000000.0d0) then
        tmp = x + (tan(y) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -9.5e-27) {
		tmp = x;
	} else if (a <= -1.55e-214) {
		tmp = x + (Math.tan(z) - a);
	} else if (a <= 195000000.0) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -9.5e-27:
		tmp = x
	elif a <= -1.55e-214:
		tmp = x + (math.tan(z) - a)
	elif a <= 195000000.0:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -9.5e-27)
		tmp = x;
	elseif (a <= -1.55e-214)
		tmp = Float64(x + Float64(tan(z) - a));
	elseif (a <= 195000000.0)
		tmp = Float64(x + Float64(tan(y) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -9.5e-27)
		tmp = x;
	elseif (a <= -1.55e-214)
		tmp = x + (tan(z) - a);
	elseif (a <= 195000000.0)
		tmp = x + (tan(y) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -9.5e-27], x, If[LessEqual[a, -1.55e-214], N[(x + N[(N[Tan[z], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 195000000.0], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.50000000000000037e-27 or 1.95e8 < a

   1. Initial program 73.7%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 23.5%

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

   if -9.50000000000000037e-27 < a < -1.55000000000000002e-214

   1. Initial program 81.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 81.4%

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

   4. Taylor expanded in y around 0 59.2%

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

   if -1.55000000000000002e-214 < a < 1.95e8

   1. Initial program 73.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 71.6%

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

   4. Taylor expanded in y around inf 58.9%

      \[\leadsto x + \left(\tan \color{blue}{y} - a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.6:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 195000000:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.6) x (if (<= a 195000000.0) (+ x (- (tan (+ y z)) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.6) {
		tmp = x;
	} else if (a <= 195000000.0) {
		tmp = x + (tan((y + z)) - a);
	} else {
		tmp = x;
	}
	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) :: tmp
    if (a <= (-1.6d0)) then
        tmp = x
    else if (a <= 195000000.0d0) then
        tmp = x + (tan((y + z)) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.6) {
		tmp = x;
	} else if (a <= 195000000.0) {
		tmp = x + (Math.tan((y + z)) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.6:
		tmp = x
	elif a <= 195000000.0:
		tmp = x + (math.tan((y + z)) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.6)
		tmp = x;
	elseif (a <= 195000000.0)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.6)
		tmp = x;
	elseif (a <= 195000000.0)
		tmp = x + (tan((y + z)) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.6], x, If[LessEqual[a, 195000000.0], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001 or 1.95e8 < a

   1. Initial program 74.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 23.3%

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

   if -1.6000000000000001 < a < 1.95e8

   1. Initial program 74.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 73.3%

      \[\leadsto x + \left(\tan \left(y + z\right) - \color{blue}{a}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 41.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 195000000:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.65) x (if (<= a 195000000.0) (+ x (- (tan y) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.65) {
		tmp = x;
	} else if (a <= 195000000.0) {
		tmp = x + (tan(y) - a);
	} else {
		tmp = x;
	}
	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) :: tmp
    if (a <= (-1.65d0)) then
        tmp = x
    else if (a <= 195000000.0d0) then
        tmp = x + (tan(y) - a)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.65) {
		tmp = x;
	} else if (a <= 195000000.0) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.65:
		tmp = x
	elif a <= 195000000.0:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.65)
		tmp = x;
	elseif (a <= 195000000.0)
		tmp = Float64(x + Float64(tan(y) - a));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (a <= -1.65)
		tmp = x;
	elseif (a <= 195000000.0)
		tmp = x + (tan(y) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.65], x, If[LessEqual[a, 195000000.0], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6499999999999999 or 1.95e8 < a

   1. Initial program 74.4%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 23.3%

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

   if -1.6499999999999999 < a < 1.95e8

   1. Initial program 74.6%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 73.3%

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

   4. Taylor expanded in y around inf 59.3%

      \[\leadsto x + \left(\tan \color{blue}{y} - a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 32.0% 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 74.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 33.1%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Alternative 14: 3.5% accurate, 207.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double a) {
	return 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 = a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in a around 0 38.6%

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

4. Taylor expanded in a around inf 3.4%

   \[\leadsto \color{blue}{-1 \cdot a} \]
5. Step-by-step derivation

   1. neg-mul-1 3.4%

      \[\leadsto \color{blue}{-a} \]

6. Simplified 3.4%

   \[\leadsto \color{blue}{-a} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 2.4%

      \[\leadsto \color{blue}{\sqrt{-a} \cdot \sqrt{-a}} \]

   2. sqrt-unprod 4.5%

      \[\leadsto \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}} \]

   3. sqr-neg 4.5%

      \[\leadsto \sqrt{\color{blue}{a \cdot a}} \]

   4. sqrt-unprod 2.4%

      \[\leadsto \color{blue}{\sqrt{a} \cdot \sqrt{a}} \]

   5. add-sqr-sqrt 3.3%

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

   6. *-un-lft-identity 3.3%

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

8. Applied egg-rr 3.3%

   \[\leadsto \color{blue}{1 \cdot a} \]
9. Step-by-step derivation

   1. *-lft-identity 3.3%

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

10. Simplified 3.3%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(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))))