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

Percentage Accurate: 96.9% → 99.8%

Time: 10.0s

Alternatives: 13

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: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z - t\right) - 1\\ t_2 := \mathsf{fma}\left(\frac{a}{t\_1}, y - z, x\right)\\ \mathbf{if}\;a \leq -6 \cdot 10^{-39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq 28500000000:\\ \;\;\;\;x - \frac{\left(z - y\right) \cdot a}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (- z t) 1.0)) (t_2 (fma (/ a t_1) (- y z) x)))
   (if (<= a -6e-39)
     t_2
     (if (<= a 28500000000.0) (- x (/ (* (- z y) a) t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) - 1.0;
	double t_2 = fma((a / t_1), (y - z), x);
	double tmp;
	if (a <= -6e-39) {
		tmp = t_2;
	} else if (a <= 28500000000.0) {
		tmp = x - (((z - y) * a) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) - 1.0)
	t_2 = fma(Float64(a / t_1), Float64(y - z), x)
	tmp = 0.0
	if (a <= -6e-39)
		tmp = t_2;
	elseif (a <= 28500000000.0)
		tmp = Float64(x - Float64(Float64(Float64(z - y) * a) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / t$95$1), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6e-39], t$95$2, If[LessEqual[a, 28500000000.0], N[(x - N[(N[(N[(z - y), $MachinePrecision] * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000055e-39 or 2.85e10 < a

   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. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

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

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lower-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 rewrites 99.9%

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

   if -6.00000000000000055e-39 < a < 2.85e10

   1. Initial program 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 100.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto x - \color{blue}{\frac{\left(y - z\right) \cdot a}{1 + \left(t - z\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{\left(z - t\right) - 1}, y - z, x\right)\\ \mathbf{elif}\;a \leq 28500000000:\\ \;\;\;\;x - \frac{\left(z - y\right) \cdot a}{\left(z - t\right) - 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{\left(z - t\right) - 1}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-y\right) \cdot a\\ t_2 := x - \frac{y - z}{\frac{\left(t - z\right) - -1}{a}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- y) a)) (t_2 (- x (/ (- y z) (/ (- (- t z) -1.0) a)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+301) (- x a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -y * a;
	double t_2 = x - ((y - z) / (((t - z) - -1.0) / a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+301) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -y * a;
	double t_2 = x - ((y - z) / (((t - z) - -1.0) / a));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 5e+301) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -y * a
	t_2 = x - ((y - z) / (((t - z) - -1.0) / a))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 5e+301:
		tmp = x - a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-y) * a)
	t_2 = Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) - -1.0) / a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+301)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = -y * a;
	t_2 = x - ((y - z) / (((t - z) - -1.0) / a));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 5e+301)
		tmp = x - a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-y) * a), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+301], N[(x - a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))) < -inf.0 or 5.0000000000000004e301 < (-.f64 x (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)))

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 88.9

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 88.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 72.2%

       \[\leadsto \left(-a\right) \cdot y \]

   if -inf.0 < (-.f64 x (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))) < 5.0000000000000004e301

   1. Initial program 95.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 66.2

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

   5. Applied rewrites 66.2%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{y - z}{\frac{\left(t - z\right) - -1}{a}} \leq -\infty:\\ \;\;\;\;\left(-y\right) \cdot a\\ \mathbf{elif}\;x - \frac{y - z}{\frac{\left(t - z\right) - -1}{a}} \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{-a}{t}, x\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8e+103)
   (fma (- y z) (/ (- a) t) x)
   (if (<= t 1.7e+94)
     (fma (/ (- z y) (- 1.0 z)) a x)
     (fma (/ z (- (- t -1.0) z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8e+103) {
		tmp = fma((y - z), (-a / t), x);
	} else if (t <= 1.7e+94) {
		tmp = fma(((z - y) / (1.0 - z)), a, x);
	} else {
		tmp = fma((z / ((t - -1.0) - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8e+103)
		tmp = fma(Float64(y - z), Float64(Float64(-a) / t), x);
	elseif (t <= 1.7e+94)
		tmp = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x);
	else
		tmp = fma(Float64(z / Float64(Float64(t - -1.0) - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8e+103], N[(N[(y - z), $MachinePrecision] * N[((-a) / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.7e+94], N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(z / N[(N[(t - -1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8e103

   1. Initial program 99.8%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-neg.f64 88.8

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

   5. Applied rewrites 88.8%

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

   7. Applied rewrites 93.3%

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

   if -8e103 < t < 1.7000000000000001e94

   1. Initial program 95.6%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 1.7000000000000001e94 < t

   1. Initial program 94.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 88.3

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

   5. Applied rewrites 88.3%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{-a}{t}, x\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- (- t -1.0) z)) a x)))
   (if (<= z -1.9e-30)
     t_1
     (if (<= z 3.3e-21) (fma (/ y (- -1.0 t)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / ((t - -1.0) - z)), a, x);
	double tmp;
	if (z <= -1.9e-30) {
		tmp = t_1;
	} else if (z <= 3.3e-21) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(Float64(t - -1.0) - z)), a, x)
	tmp = 0.0
	if (z <= -1.9e-30)
		tmp = t_1;
	elseif (z <= 3.3e-21)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(N[(t - -1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -1.9e-30], t$95$1, If[LessEqual[z, 3.3e-21], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9000000000000002e-30 or 3.30000000000000009e-21 < z

   1. Initial program 94.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 85.0

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

   5. Applied rewrites 85.0%

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

   if -1.9000000000000002e-30 < z < 3.30000000000000009e-21

   1. Initial program 98.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - -1\right) - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{a}{\left(t - z\right) - -1}, x\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ a (- (- t z) -1.0)) x)))
   (if (<= z -1.9e-30)
     t_1
     (if (<= z 3.3e-21) (fma (/ y (- -1.0 t)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (a / ((t - z) - -1.0)), x);
	double tmp;
	if (z <= -1.9e-30) {
		tmp = t_1;
	} else if (z <= 3.3e-21) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(a / Float64(Float64(t - z) - -1.0)), x)
	tmp = 0.0
	if (z <= -1.9e-30)
		tmp = t_1;
	elseif (z <= 3.3e-21)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(a / N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.9e-30], t$95$1, If[LessEqual[z, 3.3e-21], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9000000000000002e-30 or 3.30000000000000009e-21 < z

   1. Initial program 94.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 85.0

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

   5. Applied rewrites 85.0%

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

   7. Applied rewrites 80.7%

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

   if -1.9000000000000002e-30 < z < 3.30000000000000009e-21

   1. Initial program 98.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 93.1

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

   5. Applied rewrites 93.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{\left(t - z\right) - -1}, x\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{\left(t - z\right) - -1}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - (a / (((z - t) - 1.0d0) / (z - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. *-commutative N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.0

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Final simplification 99.0%

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

Alternative 7: 84.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+46}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e+46)
   (- x a)
   (if (<= z 3.3e-21) (fma (/ y (- -1.0 t)) a x) (fma (/ z (- 1.0 z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+46) {
		tmp = x - a;
	} else if (z <= 3.3e-21) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = fma((z / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e+46)
		tmp = Float64(x - a);
	elseif (z <= 3.3e-21)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e+46], N[(x - a), $MachinePrecision], If[LessEqual[z, 3.3e-21], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e46

   1. Initial program 95.3%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -2e46 < z < 3.30000000000000009e-21

   1. Initial program 99.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if 3.30000000000000009e-21 < 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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 80.1%

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

Alternative 8: 84.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+46}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2e+46)
   (- x a)
   (if (<= z 1.75e-15) (fma (/ y (- -1.0 t)) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2e+46) {
		tmp = x - a;
	} else if (z <= 1.75e-15) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2e+46)
		tmp = Float64(x - a);
	elseif (z <= 1.75e-15)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2e+46], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.75e-15], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e46 or 1.75e-15 < z

   1. Initial program 93.5%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -2e46 < z < 1.75e-15

   1. Initial program 99.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

Alternative 9: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -14:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1 - y, z, -y\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -14.0)
   (- x a)
   (if (<= z 1.75e-15) (fma (fma (- 1.0 y) z (- y)) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -14.0) {
		tmp = x - a;
	} else if (z <= 1.75e-15) {
		tmp = fma(fma((1.0 - y), z, -y), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -14.0)
		tmp = Float64(x - a);
	elseif (z <= 1.75e-15)
		tmp = fma(fma(Float64(1.0 - y), z, Float64(-y)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -14.0], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.75e-15], N[(N[(N[(1.0 - y), $MachinePrecision] * z + (-y)), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -14 or 1.75e-15 < z

   1. Initial program 93.9%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -14 < z < 1.75e-15

   1. Initial program 98.9%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.9%

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

Alternative 10: 97.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

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

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

   10. clear-num N/A

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

   11. lift-/.f64 N/A

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

   12. distribute-frac-neg2 N/A

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

   13. lower-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 rewrites 96.5%

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

5. Final simplification 96.5%

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

Alternative 11: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+46}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-y, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.75e+46) (- x a) (if (<= z 1.75e-15) (fma (- y) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.75e+46) {
		tmp = x - a;
	} else if (z <= 1.75e-15) {
		tmp = fma(-y, a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.75e+46)
		tmp = Float64(x - a);
	elseif (z <= 1.75e-15)
		tmp = fma(Float64(-y), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.75e+46], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.75e-15], N[((-y) * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.74999999999999992e46 or 1.75e-15 < z

   1. Initial program 93.5%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -1.74999999999999992e46 < z < 1.75e-15

   1. Initial program 99.0%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot y, a, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 67.1%

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

Alternative 12: 60.7% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. lower--.f64 61.8

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

5. Applied rewrites 61.8%

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

Alternative 13: 16.9% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := (-a)

Derivation

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

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

   1. lower--.f64 61.8

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

5. Applied rewrites 61.8%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 17.3%

   \[\leadsto -a \]
9. 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 2024249 
(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))))