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

Percentage Accurate: 96.9% → 99.6%

Time: 11.3s

Alternatives: 10

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: 99.6% 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 98.0%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
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. lift-/.f64 N/A

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

   6. associate-/r/ N/A

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

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

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

   8. distribute-frac-neg2 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. metadata-eval N/A

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

   15. unsub-neg N/A

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

   16. lower--.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 61.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq -5 \cdot 10^{+280}:\\ \;\;\;\;a \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- y z) (/ (+ (- t z) 1.0) a)) -5e+280) (* a (/ y z)) (- x a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((y - z) / (((t - z) + 1.0) / a)) <= -5e+280) {
		tmp = a * (y / z);
	} 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 (((y - z) / (((t - z) + 1.0d0) / a)) <= (-5d+280)) then
        tmp = a * (y / z)
    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 (((y - z) / (((t - z) + 1.0) / a)) <= -5e+280) {
		tmp = a * (y / z);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((y - z) / (((t - z) + 1.0) / a)) <= -5e+280:
		tmp = a * (y / z)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)) <= -5e+280)
		tmp = Float64(a * Float64(y / z));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((y - z) / (((t - z) + 1.0) / a)) <= -5e+280)
		tmp = a * (y / z);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], -5e+280], N[(a * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -5.0000000000000002e280

   1. Initial program 99.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-+.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 91.1

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 73.8%

       \[\leadsto a \cdot \frac{y}{\color{blue}{z}} \]

   if -5.0000000000000002e280 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 63.6

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

   5. Applied rewrites 63.6%

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

Alternative 3: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{-t}, x\right)\\ \mathbf{elif}\;t \leq 3.6:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z + -1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.8e+24)
   (fma (- y z) (/ a (- t)) x)
   (if (<= t 3.6)
     (fma (/ (- y z) (+ z -1.0)) a 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 (t <= -3.8e+24) {
		tmp = fma((y - z), (a / -t), x);
	} else if (t <= 3.6) {
		tmp = fma(((y - z) / (z + -1.0)), a, 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 (t <= -3.8e+24)
		tmp = fma(Float64(y - z), Float64(a / Float64(-t)), x);
	elseif (t <= 3.6)
		tmp = fma(Float64(Float64(y - z) / Float64(z + -1.0)), a, 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[t, -3.8e+24], N[(N[(y - z), $MachinePrecision] * N[(a / (-t)), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 3.6], N[(N[(N[(y - z), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] * a + 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 t < -3.80000000000000015e24

   1. Initial program 98.3%

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

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -3.80000000000000015e24 < t < 3.60000000000000009

   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. 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. lift-/.f64 N/A

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

      6. associate-/r/ N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. metadata-eval N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 3.60000000000000009 < t

   1. Initial program 96.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. lower-fma.f64 N/A

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

      7. lower-/.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. lower-+.f64 N/A

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

      11. lower--.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{-t}, x\right)\\ \mathbf{elif}\;t \leq 3.6:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z + -1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+136}:\\ \;\;\;\;x - y \cdot \frac{a}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ z (+ t (- 1.0 z))) x)))
   (if (<= z -4.6e+52)
     t_1
     (if (<= z 3.9e+136) (- x (* y (/ a (+ t 1.0)))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (z / (t + (1.0 - z))), x);
	double tmp;
	if (z <= -4.6e+52) {
		tmp = t_1;
	} else if (z <= 3.9e+136) {
		tmp = x - (y * (a / (t + 1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(z / Float64(t + Float64(1.0 - z))), x)
	tmp = 0.0
	if (z <= -4.6e+52)
		tmp = t_1;
	elseif (z <= 3.9e+136)
		tmp = Float64(x - Float64(y * Float64(a / Float64(t + 1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(z / N[(t + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -4.6e+52], t$95$1, If[LessEqual[z, 3.9e+136], N[(x - N[(y * N[(a / N[(t + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6e52 or 3.90000000000000019e136 < z

   1. Initial program 95.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. lower-fma.f64 N/A

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

      7. lower-/.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. lower-+.f64 N/A

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

      11. lower--.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if -4.6e52 < z < 3.90000000000000019e136

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 87.3

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

   5. Applied rewrites 87.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+136}:\\ \;\;\;\;x - y \cdot \frac{a}{t + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.9 \cdot 10^{+145}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.9e+145)
   (- x a)
   (if (<= z 4.3e+136) (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 <= -7.9e+145) {
		tmp = x - a;
	} else if (z <= 4.3e+136) {
		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 <= -7.9e+145)
		tmp = Float64(x - a);
	elseif (z <= 4.3e+136)
		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, -7.9e+145], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.3e+136], 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 < -7.8999999999999997e145 or 4.2999999999999999e136 < z

   1. Initial program 95.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -7.8999999999999997e145 < z < 4.2999999999999999e136

   1. Initial program 99.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. lower-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. lower-/.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. lower--.f64 85.8

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

   5. Applied rewrites 85.8%

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

Alternative 6: 67.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+112}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+108}:\\ \;\;\;\;x - \frac{y \cdot a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.5e+112)
   (- x a)
   (if (<= z 2.3e+108) (- x (/ (* y a) t)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.5e+112) {
		tmp = x - a;
	} else if (z <= 2.3e+108) {
		tmp = x - ((y * a) / t);
	} 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 <= (-6.5d+112)) then
        tmp = x - a
    else if (z <= 2.3d+108) then
        tmp = x - ((y * a) / t)
    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 <= -6.5e+112) {
		tmp = x - a;
	} else if (z <= 2.3e+108) {
		tmp = x - ((y * a) / t);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -6.5e+112:
		tmp = x - a
	elif z <= 2.3e+108:
		tmp = x - ((y * a) / t)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.5e+112)
		tmp = Float64(x - a);
	elseif (z <= 2.3e+108)
		tmp = Float64(x - Float64(Float64(y * a) / t));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -6.5e+112)
		tmp = x - a;
	elseif (z <= 2.3e+108)
		tmp = x - ((y * a) / t);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.5e+112], N[(x - a), $MachinePrecision], If[LessEqual[z, 2.3e+108], N[(x - N[(N[(y * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999998e112 or 2.2999999999999999e108 < z

   1. Initial program 95.5%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if -6.4999999999999998e112 < z < 2.2999999999999999e108

   1. Initial program 99.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+112}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+108}:\\ \;\;\;\;x - \frac{y \cdot a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{z}{t}, x\right)\\ \mathbf{if}\;t \leq -8.6 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+95}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ z t) x)))
   (if (<= t -8.6e+35) t_1 (if (<= t 5.8e+95) (- x a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (z / t), x);
	double tmp;
	if (t <= -8.6e+35) {
		tmp = t_1;
	} else if (t <= 5.8e+95) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(z / t), x)
	tmp = 0.0
	if (t <= -8.6e+35)
		tmp = t_1;
	elseif (t <= 5.8e+95)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -8.6e+35], t$95$1, If[LessEqual[t, 5.8e+95], N[(x - a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.5999999999999995e35 or 5.80000000000000027e95 < t

   1. Initial program 97.0%

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

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 71.5%

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

   if -8.5999999999999995e35 < t < 5.80000000000000027e95

   1. Initial program 98.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 67.6

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

   5. Applied rewrites 67.6%

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

Alternative 8: 97.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
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 98.3%

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

5. Final simplification 98.3%

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

Alternative 9: 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 98.0%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 61.4

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

5. Applied rewrites 61.4%

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

Alternative 10: 17.0% 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 98.0%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 61.4

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

5. Applied rewrites 61.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 16.2%

   \[\leadsto -a \]
9. Add Preprocessing

Developer Target 1: 99.6% 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 2024221 
(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))))