Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 96.9% → 99.7%

Time: 12.5s

Alternatives: 14

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.7%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. associate-/r/ N/A

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

   4. distribute-lft-neg-in N/A

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

   5. distribute-frac-neg2 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. --lowering--.f64 N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-in N/A

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

   11. metadata-eval N/A

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

   12. unsub-neg N/A

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

   13. --lowering--.f64 N/A

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

   14. --lowering--.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 57.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+91}:\\ \;\;\;\;-a\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+194}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_1 -5e+91) (- a) (if (<= t_1 5e+194) x (- a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -5e+91) {
		tmp = -a;
	} else if (t_1 <= 5e+194) {
		tmp = x;
	} else {
		tmp = -a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - z) / (((t - z) + 1.0d0) / a)
    if (t_1 <= (-5d+91)) then
        tmp = -a
    else if (t_1 <= 5d+194) then
        tmp = x
    else
        tmp = -a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -5e+91) {
		tmp = -a;
	} else if (t_1 <= 5e+194) {
		tmp = x;
	} else {
		tmp = -a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_1 <= -5e+91:
		tmp = -a
	elif t_1 <= 5e+194:
		tmp = x
	else:
		tmp = -a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_1 <= -5e+91)
		tmp = Float64(-a);
	elseif (t_1 <= 5e+194)
		tmp = x;
	else
		tmp = Float64(-a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_1 <= -5e+91)
		tmp = -a;
	elseif (t_1 <= 5e+194)
		tmp = x;
	else
		tmp = -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+91], (-a), If[LessEqual[t$95$1, 5e+194], x, (-a)]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -5.0000000000000002e91 or 4.99999999999999989e194 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 39.5%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(a\right)} \]

      2. neg-lowering-neg.f64 31.9

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

   8. Simplified 31.9%

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

   if -5.0000000000000002e91 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 4.99999999999999989e194

   1. Initial program 95.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 67.3%

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

Alternative 3: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \mathbf{elif}\;z \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{-1 - t}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e+36)
   (fma (/ (- y z) z) a x)
   (if (<= z 1.2)
     (fma (/ a (- -1.0 t)) (- y z) x)
     (fma a (/ z (+ t (- 1.0 z))) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e+36) {
		tmp = fma(((y - z) / z), a, x);
	} else if (z <= 1.2) {
		tmp = fma((a / (-1.0 - t)), (y - z), x);
	} else {
		tmp = fma(a, (z / (t + (1.0 - z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e+36)
		tmp = fma(Float64(Float64(y - z) / z), a, x);
	elseif (z <= 1.2)
		tmp = fma(Float64(a / Float64(-1.0 - t)), Float64(y - z), x);
	else
		tmp = fma(a, Float64(z / Float64(t + Float64(1.0 - z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e+36], N[(N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 1.2], N[(N[(a / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(z / N[(t + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3000000000000001e36

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 87.4%

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

   if -1.3000000000000001e36 < z < 1.19999999999999996

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. --lowering--.f64 97.9

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

   7. Simplified 97.9%

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

   if 1.19999999999999996 < z

   1. Initial program 92.4%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 87.4

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

   5. Simplified 87.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{z + -1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.1e+32)
   (fma a (/ y (- -1.0 t)) x)
   (if (<= t 3.6e+144)
     (fma (- y z) (/ a (+ z -1.0)) x)
     (fma (/ (- z y) t) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.1e+32) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else if (t <= 3.6e+144) {
		tmp = fma((y - z), (a / (z + -1.0)), x);
	} else {
		tmp = fma(((z - y) / t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.1e+32)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	elseif (t <= 3.6e+144)
		tmp = fma(Float64(y - z), Float64(a / Float64(z + -1.0)), x);
	else
		tmp = fma(Float64(Float64(z - y) / t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.1e+32], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 3.6e+144], N[(N[(y - z), $MachinePrecision] * N[(a / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.09999999999999993e32

   1. Initial program 93.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right) + x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{y}{1 + t}}\right)\right) + x \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{y}{1 + t}\right)\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 85.6

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

   5. Simplified 85.6%

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

   if -3.09999999999999993e32 < t < 3.5999999999999997e144

   1. Initial program 97.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-in N/A

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

      9. mul-1-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. mul-1-neg N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 94.9%

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

   if 3.5999999999999997e144 < t

   1. Initial program 99.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(\left(y - z\right)\right)}}{t}, a, x\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)}\right)}{t}, a, x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)}\right)}{t}, a, x\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}}{t}, a, x\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y}}{t}, a, x\right) \]

      8. remove-double-neg N/A

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

      9. --lowering--.f64 89.7

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

   7. Simplified 89.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{z + -1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.85e+38)
   (fma (/ (- y z) z) a x)
   (if (<= z 7.2e-75)
     (fma a (/ y (- -1.0 t)) x)
     (fma a (/ z (+ t (- 1.0 z))) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.85e+38) {
		tmp = fma(((y - z) / z), a, x);
	} else if (z <= 7.2e-75) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = fma(a, (z / (t + (1.0 - z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.85e+38)
		tmp = fma(Float64(Float64(y - z) / z), a, x);
	elseif (z <= 7.2e-75)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = fma(a, Float64(z / Float64(t + Float64(1.0 - z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.85e+38], N[(N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 7.2e-75], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(z / N[(t + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8500000000000001e38

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 87.4%

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

   if -1.8500000000000001e38 < z < 7.2000000000000001e-75

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right) + x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{y}{1 + t}}\right)\right) + x \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{y}{1 + t}\right)\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 93.8

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

   5. Simplified 93.8%

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

   if 7.2000000000000001e-75 < z

   1. Initial program 93.4%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 87.9

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

   5. Simplified 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.4e+35)
   (fma (/ (- y z) z) a x)
   (if (<= z 1.0) (fma a (/ y (- -1.0 t)) x) (fma a (/ z (- t z)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.4e+35) {
		tmp = fma(((y - z) / z), a, x);
	} else if (z <= 1.0) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = fma(a, (z / (t - z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.4e+35)
		tmp = fma(Float64(Float64(y - z) / z), a, x);
	elseif (z <= 1.0)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = fma(a, Float64(z / Float64(t - z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.4e+35], N[(N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 1.0], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.40000000000000005e35

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 87.4%

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

   if -5.40000000000000005e35 < z < 1

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right) + x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{y}{1 + t}}\right)\right) + x \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{y}{1 + t}\right)\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 93.6

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

   5. Simplified 93.6%

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

   if 1 < z

   1. Initial program 92.4%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. --lowering--.f64 87.4

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

   5. Simplified 87.4%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(a, \frac{z}{t + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}, x\right) \]

      2. neg-lowering-neg.f64 86.8

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

   8. Simplified 86.8%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 98000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) z) a x)))
   (if (<= z -1.6e+37)
     t_1
     (if (<= z 98000.0) (fma a (/ y (- -1.0 t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / z), a, x);
	double tmp;
	if (z <= -1.6e+37) {
		tmp = t_1;
	} else if (z <= 98000.0) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / z), a, x)
	tmp = 0.0
	if (z <= -1.6e+37)
		tmp = t_1;
	elseif (z <= 98000.0)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -1.6e+37], t$95$1, If[LessEqual[z, 98000.0], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000007e37 or 98000 < z

   1. Initial program 94.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 86.0%

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

   if -1.60000000000000007e37 < z < 98000

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right) + x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{y}{1 + t}}\right)\right) + x \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{y}{1 + t}\right)\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 93.6

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

   5. Simplified 93.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+39}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e+39)
   (- x a)
   (if (<= z 1e+94) (fma a (/ y (- -1.0 t)) x) (fma a (/ z (- 1.0 z)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e+39) {
		tmp = x - a;
	} else if (z <= 1e+94) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = fma(a, (z / (1.0 - z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.8e+39)
		tmp = Float64(x - a);
	elseif (z <= 1e+94)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = fma(a, Float64(z / Float64(1.0 - z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.8e+39], N[(x - a), $MachinePrecision], If[LessEqual[z, 1e+94], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.80000000000000001e39

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 73.8%

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

   if -2.80000000000000001e39 < z < 1e94

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right) + x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{y}{1 + t}}\right)\right) + x \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{y}{1 + t}\right)\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 89.2

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

   5. Simplified 89.2%

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

   if 1e94 < z

   1. Initial program 89.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 63.6

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

   5. Simplified 63.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x - -1 \cdot \frac{a \cdot z}{1 - z}} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 84.5

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

   8. Simplified 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+36}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.1e+36)
   (- x a)
   (if (<= z 1.35e+94) (fma a (/ y (- -1.0 t)) x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e+36) {
		tmp = x - a;
	} else if (z <= 1.35e+94) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.1e+36)
		tmp = Float64(x - a);
	elseif (z <= 1.35e+94)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.1e+36], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.35e+94], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000004e36 or 1.3500000000000001e94 < z

   1. Initial program 93.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 78.9%

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

   if -2.10000000000000004e36 < z < 1.3500000000000001e94

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{1 + t}\right)\right) + x} \]

      3. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{a \cdot \frac{y}{1 + t}}\right)\right) + x \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{a \cdot \left(\mathsf{neg}\left(\frac{y}{1 + t}\right)\right)} + x \]

      5. accelerator-lowering-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 89.2

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

   5. Simplified 89.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{-1 + \left(z - t\right)}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 6.6e+153)
   (fma (/ a (+ -1.0 (- z t))) (- y z) x)
   (fma (/ (- y z) z) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 6.6e+153) {
		tmp = fma((a / (-1.0 + (z - t))), (y - z), x);
	} else {
		tmp = fma(((y - z) / z), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 6.6e+153)
		tmp = fma(Float64(a / Float64(-1.0 + Float64(z - t))), Float64(y - z), x);
	else
		tmp = fma(Float64(Float64(y - z) / z), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 6.6e+153], N[(N[(a / N[(-1.0 + N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / z), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.59999999999999989e153

   1. Initial program 98.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. distribute-lft-neg-in N/A

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

      7. clear-num N/A

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

      8. distribute-frac-neg2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 98.6%

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

   if 6.59999999999999989e153 < z

   1. Initial program 86.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/r/ N/A

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

      4. distribute-lft-neg-in N/A

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

      5. distribute-frac-neg2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. --lowering--.f64 N/A

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

      14. --lowering--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 100.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{-1 + \left(z - t\right)}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - \frac{a}{z}\right) - a\\ \mathbf{if}\;z \leq -6 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.9:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (- x (/ a z)) a)))
   (if (<= z -6e-17) t_1 (if (<= z 0.9) (- x (* y a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (a / z)) - a;
	double tmp;
	if (z <= -6e-17) {
		tmp = t_1;
	} else if (z <= 0.9) {
		tmp = x - (y * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - (a / z)) - a
    if (z <= (-6d-17)) then
        tmp = t_1
    else if (z <= 0.9d0) then
        tmp = x - (y * a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (a / z)) - a;
	double tmp;
	if (z <= -6e-17) {
		tmp = t_1;
	} else if (z <= 0.9) {
		tmp = x - (y * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (a / z)) - a
	tmp = 0
	if z <= -6e-17:
		tmp = t_1
	elif z <= 0.9:
		tmp = x - (y * a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(a / z)) - a)
	tmp = 0.0
	if (z <= -6e-17)
		tmp = t_1;
	elseif (z <= 0.9)
		tmp = Float64(x - Float64(y * a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (a / z)) - a;
	tmp = 0.0;
	if (z <= -6e-17)
		tmp = t_1;
	elseif (z <= 0.9)
		tmp = x - (y * a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(a / z), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[z, -6e-17], t$95$1, If[LessEqual[z, 0.9], N[(x - N[(y * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.00000000000000012e-17 or 0.900000000000000022 < z

   1. Initial program 94.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 63.6

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

   5. Simplified 63.6%

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

   6. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. distribute-lft-out-- N/A

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

      8. associate-*r/ N/A

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

      9. associate--l+ N/A

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

      10. --lowering--.f64 N/A

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

   8. Simplified 77.1%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{a}{z}\right)} - a \]
   10. Step-by-step derivation

       1. --lowering--.f64 N/A

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

       2. /-lowering-/.f64 74.3

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

   11. Simplified 74.3%

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

   if -6.00000000000000012e-17 < z < 0.900000000000000022

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 77.7

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

   5. Simplified 77.7%

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

   6. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - a \cdot y} \]

      2. *-lowering-*.f64 73.3

         \[\leadsto x - \color{blue}{a \cdot y} \]

   8. Simplified 73.3%

      \[\leadsto \color{blue}{x - a \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-17}:\\ \;\;\;\;\left(x - \frac{a}{z}\right) - a\\ \mathbf{elif}\;z \leq 0.9:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{a}{z}\right) - a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-17}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.02:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6e-17) (- x a) (if (<= z 1.02) (- x (* y a)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e-17) {
		tmp = x - a;
	} else if (z <= 1.02) {
		tmp = x - (y * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-6d-17)) then
        tmp = x - a
    else if (z <= 1.02d0) then
        tmp = x - (y * a)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6e-17) {
		tmp = x - a;
	} else if (z <= 1.02) {
		tmp = x - (y * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6e-17:
		tmp = x - a
	elif z <= 1.02:
		tmp = x - (y * a)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6e-17)
		tmp = Float64(x - a);
	elseif (z <= 1.02)
		tmp = Float64(x - Float64(y * a));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6e-17)
		tmp = x - a;
	elseif (z <= 1.02)
		tmp = x - (y * a);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6e-17], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.02], N[(x - N[(y * a), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.00000000000000012e-17 or 1.02 < z

   1. Initial program 94.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 73.8%

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

   if -6.00000000000000012e-17 < z < 1.02

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 77.7

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

   5. Simplified 77.7%

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

   6. Taylor expanded in z around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \color{blue}{x - a \cdot y} \]

      2. *-lowering-*.f64 73.3

         \[\leadsto x - \color{blue}{a \cdot y} \]

   8. Simplified 73.3%

      \[\leadsto \color{blue}{x - a \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-17}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.02:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-95}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.5e-95) (- x a) (if (<= z 9.5e+93) x (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-95) {
		tmp = x - a;
	} else if (z <= 9.5e+93) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-8.5d-95)) then
        tmp = x - a
    else if (z <= 9.5d+93) then
        tmp = x
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.5e-95) {
		tmp = x - a;
	} else if (z <= 9.5e+93) {
		tmp = x;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.5e-95:
		tmp = x - a
	elif z <= 9.5e+93:
		tmp = x
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.5e-95)
		tmp = Float64(x - a);
	elseif (z <= 9.5e+93)
		tmp = x;
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -8.5e-95)
		tmp = x - a;
	elseif (z <= 9.5e+93)
		tmp = x;
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.5e-95], N[(x - a), $MachinePrecision], If[LessEqual[z, 9.5e+93], x, N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.4999999999999995e-95 or 9.4999999999999991e93 < z

   1. Initial program 94.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Simplified 72.2%

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

   if -8.4999999999999995e-95 < z < 9.4999999999999991e93

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 57.7%

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

Alternative 14: 54.0% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.7%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 50.7%

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

Developer Target 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (/ (- y z) (+ (- t z) 1)) a)))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))