tan-example (used to crash)

Percentage Accurate: 79.4% → 99.7%

Time: 40.1s

Alternatives: 11

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: 79.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. +-commutative 80.9%

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

   2. sub-neg 80.9%

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

   3. associate-+l+ 80.9%

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

   4. tan-sum 99.6%

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

   5. div-inv 99.6%

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

   6. fma-define 99.6%

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

   7. neg-mul-1 99.6%

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

   8. fma-define 99.6%

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

4. Applied egg-rr 99.6%

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

   1. fma-undefine 99.6%

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

   2. fma-undefine 99.6%

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

   3. neg-mul-1 99.6%

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

   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. unsub-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. Add Preprocessing

Alternative 2: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\cos \left(y + z\right)}\\ t_1 := \sin \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \mathsf{fma}\left(t\_1, t\_0, -\tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0001:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_1 \cdot t\_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (cos (+ y z)))) (t_1 (sin (+ y z))))
   (if (<= (tan a) -0.01)
     (+ x (fma t_1 t_0 (- (tan a))))
     (if (<= (tan a) 0.0001)
       (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))
       (+ x (- (* t_1 t_0) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 / cos((y + z));
	double t_1 = sin((y + z));
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = x + fma(t_1, t_0, -tan(a));
	} else if (tan(a) <= 0.0001) {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - a);
	} else {
		tmp = x + ((t_1 * t_0) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 / cos(Float64(y + z)))
	t_1 = sin(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = Float64(x + fma(t_1, t_0, Float64(-tan(a))));
	elseif (tan(a) <= 0.0001)
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - a));
	else
		tmp = Float64(x + Float64(Float64(t_1 * t_0) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[(x + N[(t$95$1 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 0.0001], 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] - a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(t$95$1 * t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0100000000000000002

   1. Initial program 79.8%

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

      1. tan-quot 79.8%

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

      2. div-inv 79.8%

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

      3. fma-neg 79.8%

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

   4. Applied egg-rr 79.8%

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

   if -0.0100000000000000002 < (tan.f64 a) < 1.00000000000000005e-4

   1. Initial program 83.9%

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

   3. Taylor expanded in a around 0 83.9%

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

      1. tan-sum 99.8%

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

      2. tan-quot 99.8%

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

      3. div-inv 99.8%

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

      4. tan-quot 99.8%

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

      5. fma-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. *-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. fma-undefine 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   if 1.00000000000000005e-4 < (tan.f64 a)

   1. Initial program 76.1%

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

      1. tan-quot 76.1%

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

      2. div-inv 76.1%

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

   4. Applied egg-rr 76.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, -\tan a\right)\\ \mathbf{elif}\;\tan a \leq 0.0001:\\ \;\;\;\;x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\cos \left(y + z\right)}\\ t_1 := \sin \left(y + z\right)\\ \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \mathsf{fma}\left(t\_1, t\_0, -\tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_1 \cdot t\_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (/ 1.0 (cos (+ y z)))) (t_1 (sin (+ y z))))
   (if (<= (tan a) -0.01)
     (+ x (fma t_1 t_0 (- (tan a))))
     (if (<= (tan a) 2e-19)
       (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))))
       (+ x (- (* t_1 t_0) (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = 1.0 / cos((y + z));
	double t_1 = sin((y + z));
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = x + fma(t_1, t_0, -tan(a));
	} else if (tan(a) <= 2e-19) {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + ((t_1 * t_0) - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(1.0 / cos(Float64(y + z)))
	t_1 = sin(Float64(y + z))
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = Float64(x + fma(t_1, t_0, Float64(-tan(a))));
	elseif (tan(a) <= 2e-19)
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(Float64(t_1 * t_0) - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[(x + N[(t$95$1 * t$95$0 + (-N[Tan[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-19], 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[(N[(t$95$1 * t$95$0), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0100000000000000002

   1. Initial program 79.8%

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

      1. tan-quot 79.8%

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

      2. div-inv 79.8%

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

      3. fma-neg 79.8%

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

   4. Applied egg-rr 79.8%

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

   if -0.0100000000000000002 < (tan.f64 a) < 2e-19

   1. Initial program 84.1%

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

      1. tan-quot 84.1%

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

      2. div-inv 84.1%

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

   4. Applied egg-rr 84.1%

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

   5. Taylor expanded in a around 0 84.1%

      \[\leadsto \color{blue}{x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 84.1%

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

      2. remove-double-neg 84.1%

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

      3. mul-1-neg 84.1%

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

      4. sub-neg 84.1%

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

      5. remove-double-neg 84.1%

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

      6. mul-1-neg 84.1%

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

      7. sub-neg 84.1%

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

   7. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} + x} \]
   8. Step-by-step derivation

      1. quot-tan 84.1%

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

      2. tan-sum 99.8%

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

      3. div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. *-rgt-identity 99.8%

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

   11. Simplified 99.8%

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

   if 2e-19 < (tan.f64 a)

   1. Initial program 76.1%

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

      1. tan-quot 76.1%

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

      2. div-inv 76.1%

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

   4. Applied egg-rr 76.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \mathsf{fma}\left(\sin \left(y + z\right), \frac{1}{\cos \left(y + z\right)}, -\tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)} - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (tan a) -0.01)
   (+ x (- (tan (+ y z)) (tan a)))
   (if (<= (tan a) 2e-19)
     (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))))
     (+ x (- (* (sin (+ y z)) (/ 1.0 (cos (+ y z)))) (tan a))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (tan(a) <= -0.01) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 2e-19) {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	} else {
		tmp = x + ((sin((y + z)) * (1.0 / cos((y + 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 (tan(a) <= (-0.01d0)) then
        tmp = x + (tan((y + z)) - tan(a))
    else if (tan(a) <= 2d-19) then
        tmp = x + ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z))))
    else
        tmp = x + ((sin((y + z)) * (1.0d0 / cos((y + 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 (Math.tan(a) <= -0.01) {
		tmp = x + (Math.tan((y + z)) - Math.tan(a));
	} else if (Math.tan(a) <= 2e-19) {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z))));
	} else {
		tmp = x + ((Math.sin((y + z)) * (1.0 / Math.cos((y + z)))) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if math.tan(a) <= -0.01:
		tmp = x + (math.tan((y + z)) - math.tan(a))
	elif math.tan(a) <= 2e-19:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z))))
	else:
		tmp = x + ((math.sin((y + z)) * (1.0 / math.cos((y + z)))) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (tan(a) <= -0.01)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 2e-19)
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))));
	else
		tmp = Float64(x + Float64(Float64(sin(Float64(y + z)) * Float64(1.0 / cos(Float64(y + z)))) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (tan(a) <= -0.01)
		tmp = x + (tan((y + z)) - tan(a));
	elseif (tan(a) <= 2e-19)
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	else
		tmp = x + ((sin((y + z)) * (1.0 / cos((y + z)))) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[Tan[a], $MachinePrecision], -0.01], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-19], 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[(N[(N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -0.0100000000000000002

   1. Initial program 79.8%

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

   if -0.0100000000000000002 < (tan.f64 a) < 2e-19

   1. Initial program 84.1%

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

      1. tan-quot 84.1%

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

      2. div-inv 84.1%

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

   4. Applied egg-rr 84.1%

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

   5. Taylor expanded in a around 0 84.1%

      \[\leadsto \color{blue}{x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 84.1%

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

      2. remove-double-neg 84.1%

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

      3. mul-1-neg 84.1%

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

      4. sub-neg 84.1%

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

      5. remove-double-neg 84.1%

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

      6. mul-1-neg 84.1%

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

      7. sub-neg 84.1%

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

   7. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} + x} \]
   8. Step-by-step derivation

      1. quot-tan 84.1%

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

      2. tan-sum 99.8%

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

      3. div-inv 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.8%

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

       2. *-rgt-identity 99.8%

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

   11. Simplified 99.8%

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

   if 2e-19 < (tan.f64 a)

   1. Initial program 76.1%

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

      1. tan-quot 76.1%

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

      2. div-inv 76.1%

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

   4. Applied egg-rr 76.1%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.01:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\sin \left(y + z\right) \cdot \frac{1}{\cos \left(y + z\right)} - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 1e-27) (+ x (- (tan y) (tan a))) (+ x (tan (+ y z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-27) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + tan((y + z));
	}
	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) <= 1d-27) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + tan((y + z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 1e-27) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + Math.tan((y + z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 1e-27:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + math.tan((y + z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 1e-27)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + tan(Float64(y + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 1e-27)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + tan((y + z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 1e-27], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 1e-27

   1. Initial program 86.4%

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

   3. Taylor expanded in y around inf 68.0%

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

   if 1e-27 < (+.f64 y z)

   1. Initial program 69.9%

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

      1. tan-quot 69.9%

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

      2. div-inv 69.9%

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in a around 0 50.9%

      \[\leadsto \color{blue}{x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 50.9%

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

      2. remove-double-neg 50.9%

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

      3. mul-1-neg 50.9%

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

      4. sub-neg 50.9%

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

      5. remove-double-neg 50.9%

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

      6. mul-1-neg 50.9%

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

      7. sub-neg 50.9%

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

   7. Simplified 50.9%

      \[\leadsto \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} + x} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 50.9%

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

      2. quot-tan 50.9%

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

   9. Applied egg-rr 50.9%

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

       1. *-lft-identity 50.9%

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

   11. Simplified 50.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 10^{-27}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= y -3.6e-7) (+ 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 <= -3.6e-7) {
		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 <= (-3.6d-7)) 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 <= -3.6e-7) {
		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 <= -3.6e-7:
		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 (y <= -3.6e-7)
		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 <= -3.6e-7)
		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[y, -3.6e-7], 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 y < -3.59999999999999994e-7

   1. Initial program 65.1%

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

   3. Taylor expanded in y around inf 64.3%

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

   if -3.59999999999999994e-7 < y

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 0 77.1%

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

Alternative 7: 79.4% 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 80.9%

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

Alternative 8: 41.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.55:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;x + \left(\tan y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= a -1.55) x (if (<= a 5.6e-5) (+ x (- (tan y) a)) x)))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (a <= -1.55) {
		tmp = x;
	} else if (a <= 5.6e-5) {
		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 <= (-1.55d0)) then
        tmp = x
    else if (a <= 5.6d-5) 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 <= -1.55) {
		tmp = x;
	} else if (a <= 5.6e-5) {
		tmp = x + (Math.tan(y) - a);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if a <= -1.55:
		tmp = x
	elif a <= 5.6e-5:
		tmp = x + (math.tan(y) - a)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (a <= -1.55)
		tmp = x;
	elseif (a <= 5.6e-5)
		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 <= -1.55)
		tmp = x;
	elseif (a <= 5.6e-5)
		tmp = x + (tan(y) - a);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[a, -1.55], x, If[LessEqual[a, 5.6e-5], N[(x + N[(N[Tan[y], $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.55000000000000004 or 5.59999999999999992e-5 < a

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf 22.6%

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

   if -1.55000000000000004 < a < 5.59999999999999992e-5

   1. Initial program 84.6%

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

   3. Taylor expanded in a around 0 83.8%

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

   4. Taylor expanded in y around inf 65.4%

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

Alternative 9: 50.4% 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 80.9%

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

   1. tan-quot 80.9%

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

   2. div-inv 80.9%

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

4. Applied egg-rr 80.9%

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

5. Taylor expanded in a around 0 53.5%

   \[\leadsto \color{blue}{x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} \]
6. Step-by-step derivation

   1. +-commutative 53.5%

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

   2. remove-double-neg 53.5%

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

   3. mul-1-neg 53.5%

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

   4. sub-neg 53.5%

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

   5. remove-double-neg 53.5%

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

   6. mul-1-neg 53.5%

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

   7. sub-neg 53.5%

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

7. Simplified 53.5%

   \[\leadsto \color{blue}{\frac{\sin \left(z + y\right)}{\cos \left(z + y\right)} + x} \]
8. Step-by-step derivation

   1. *-un-lft-identity 53.5%

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

   2. quot-tan 53.5%

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

9. Applied egg-rr 53.5%

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

    1. *-lft-identity 53.5%

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

11. Simplified 53.5%

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

12. Final simplification 53.5%

    \[\leadsto x + \tan \left(y + z\right) \]
13. Add Preprocessing

Alternative 10: 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 80.9%

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

3. Taylor expanded in x around inf 31.6%

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

Alternative 11: 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 80.9%

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

3. Taylor expanded in a around 0 44.2%

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

4. Taylor expanded in a around inf 3.6%

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

   1. neg-mul-1 3.6%

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

6. Simplified 3.6%

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

   1. add-sqr-sqrt 2.6%

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

   2. sqrt-unprod 5.1%

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

   3. sqr-neg 5.1%

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

   4. sqrt-unprod 2.7%

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

   5. add-sqr-sqrt 3.7%

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

   6. *-un-lft-identity 3.7%

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

8. Applied egg-rr 3.7%

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

   1. *-lft-identity 3.7%

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

10. Simplified 3.7%

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

Reproduce

herbie shell --seed 2024146 
(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))))