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

Percentage Accurate: 97.3% → 99.6%

Time: 12.2s

Alternatives: 15

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: 97.3% 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 95.2%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. frac-2neg N/A

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

   5. frac-2neg N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-/r/ N/A

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

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

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

   14. distribute-frac-neg2 N/A

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

   15. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{-a}{t}, x\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z + -1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- a) t) x)))
   (if (<= t -4.5e+89)
     t_1
     (if (<= t 6.4e+41) (fma (/ (- y z) (+ z -1.0)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (-a / t), x);
	double tmp;
	if (t <= -4.5e+89) {
		tmp = t_1;
	} else if (t <= 6.4e+41) {
		tmp = fma(((y - z) / (z + -1.0)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(-a) / t), x)
	tmp = 0.0
	if (t <= -4.5e+89)
		tmp = t_1;
	elseif (t <= 6.4e+41)
		tmp = fma(Float64(Float64(y - z) / Float64(z + -1.0)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[((-a) / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -4.5e+89], t$95$1, If[LessEqual[t, 6.4e+41], N[(N[(N[(y - z), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.5e89 or 6.40000000000000019e41 < t

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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. *-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 86.4

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

   5. Applied rewrites 86.4%

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

   if -4.5e89 < t < 6.40000000000000019e41

   1. Initial program 93.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.2

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

   7. Applied rewrites 96.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{-a}{t}, x\right)\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z + -1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{-a}{t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{elif}\;z \leq 130000000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, y - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e+80)
   (fma a (/ z (- t z)) x)
   (if (<= z 130000000.0) (fma a (/ y (- -1.0 t)) x) (fma (/ a z) (- y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+80) {
		tmp = fma(a, (z / (t - z)), x);
	} else if (z <= 130000000.0) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = fma((a / z), (y - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e+80)
		tmp = fma(a, Float64(z / Float64(t - z)), x);
	elseif (z <= 130000000.0)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = fma(Float64(a / z), Float64(y - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e+80], N[(a * N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 130000000.0], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(a / z), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e80

   1. Initial program 93.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if -1e80 < z < 1.3e8

   1. Initial program 98.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

   if 1.3e8 < z

   1. Initial program 87.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 80.7

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

   7. Applied rewrites 80.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t - z}, x\right)\\ \mathbf{elif}\;z \leq 130000000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a}{z}, y - z, x\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 130000000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ a z) (- y z) x)))
   (if (<= z -2.5e+76)
     t_1
     (if (<= z 130000000.0) (fma a (/ y (- -1.0 t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((a / z), (y - z), x);
	double tmp;
	if (z <= -2.5e+76) {
		tmp = t_1;
	} else if (z <= 130000000.0) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(a / z), Float64(y - z), x)
	tmp = 0.0
	if (z <= -2.5e+76)
		tmp = t_1;
	elseif (z <= 130000000.0)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a / z), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.5e+76], t$95$1, If[LessEqual[z, 130000000.0], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999996e76 or 1.3e8 < z

   1. Initial program 90.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 80.5

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

   7. Applied rewrites 80.5%

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

   if -2.49999999999999996e76 < z < 1.3e8

   1. Initial program 98.2%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. 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 92.0

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

   5. Applied rewrites 92.0%

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

Alternative 5: 84.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+118}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 240000000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e+118)
   (- x a)
   (if (<= z 240000000.0) (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 <= -1.6e+118) {
		tmp = x - a;
	} else if (z <= 240000000.0) {
		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 <= -1.6e+118)
		tmp = Float64(x - a);
	elseif (z <= 240000000.0)
		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, -1.6e+118], N[(x - a), $MachinePrecision], If[LessEqual[z, 240000000.0], 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 < -1.60000000000000008e118 or 2.4e8 < z

   1. Initial program 89.3%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -1.60000000000000008e118 < z < 2.4e8

   1. Initial program 98.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 90.2

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

   5. Applied rewrites 90.2%

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

Alternative 6: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{-1 + \left(z - t\right)}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 1.85e+161)
   (fma (/ a (+ -1.0 (- z t))) (- y z) x)
   (fma (* (- y z) (/ 1.0 z)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.85e+161) {
		tmp = fma((a / (-1.0 + (z - t))), (y - z), x);
	} else {
		tmp = fma(((y - z) * (1.0 / z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 1.85e+161)
		tmp = fma(Float64(a / Float64(-1.0 + Float64(z - t))), Float64(y - z), x);
	else
		tmp = fma(Float64(Float64(y - z) * Float64(1.0 / z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 1.85e+161], N[(N[(a / N[(-1.0 + N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.8499999999999999e161

   1. Initial program 96.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

   4. Applied rewrites 97.2%

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

   if 1.8499999999999999e161 < z

   1. Initial program 76.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.85 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{-1 + \left(z - t\right)}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y - z\right) \cdot \frac{1}{z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y}{-t}, x\right)\\ \mathbf{if}\;t \leq -6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(t, a, -a\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ y (- t)) x)))
   (if (<= t -6.0) t_1 (if (<= t 3.1e-19) (fma y (fma t a (- a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (y / -t), x);
	double tmp;
	if (t <= -6.0) {
		tmp = t_1;
	} else if (t <= 3.1e-19) {
		tmp = fma(y, fma(t, a, -a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(y / Float64(-t)), x)
	tmp = 0.0
	if (t <= -6.0)
		tmp = t_1;
	elseif (t <= 3.1e-19)
		tmp = fma(y, fma(t, a, Float64(-a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(y / (-t)), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -6.0], t$95$1, If[LessEqual[t, 3.1e-19], N[(y * N[(t * a + (-a)), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6 or 3.0999999999999999e-19 < t

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 77.2

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

   8. Applied rewrites 77.2%

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

   if -6 < t < 3.0999999999999999e-19

   1. Initial program 94.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 69.2

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

   7. Applied rewrites 69.2%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 69.2

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

   10. Applied rewrites 69.2%

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

Alternative 8: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot \frac{a}{t}\\ \mathbf{if}\;t \leq -6:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(t, a, -a\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y (/ a t)))))
   (if (<= t -6.0) t_1 (if (<= t 3.1e-19) (fma y (fma t a (- a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * (a / t));
	double tmp;
	if (t <= -6.0) {
		tmp = t_1;
	} else if (t <= 3.1e-19) {
		tmp = fma(y, fma(t, a, -a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * Float64(a / t)))
	tmp = 0.0
	if (t <= -6.0)
		tmp = t_1;
	elseif (t <= 3.1e-19)
		tmp = fma(y, fma(t, a, Float64(-a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.0], t$95$1, If[LessEqual[t, 3.1e-19], N[(y * N[(t * a + (-a)), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6 or 3.0999999999999999e-19 < t

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   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 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 71.2

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

   8. Applied rewrites 71.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 71.2

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 75.5

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

   10. Applied rewrites 75.5%

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

   if -6 < t < 3.0999999999999999e-19

   1. Initial program 94.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 69.2

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

   7. Applied rewrites 69.2%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 69.2

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

   10. Applied rewrites 69.2%

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

Alternative 9: 75.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.72:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 1 - y, -y\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.72)
   (- x a)
   (if (<= z 4.6e-13) (fma (fma z (- 1.0 y) (- y)) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.72) {
		tmp = x - a;
	} else if (z <= 4.6e-13) {
		tmp = fma(fma(z, (1.0 - y), -y), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.72)
		tmp = Float64(x - a);
	elseif (z <= 4.6e-13)
		tmp = fma(fma(z, Float64(1.0 - y), Float64(-y)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.72], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.6e-13], N[(N[(z * N[(1.0 - y), $MachinePrecision] + (-y)), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.71999999999999997 or 4.59999999999999958e-13 < z

   1. Initial program 92.1%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -0.71999999999999997 < z < 4.59999999999999958e-13

   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

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 71.4

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

   10. Applied rewrites 71.4%

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

Alternative 10: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -66:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, 1, -y\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -66.0)
   (- x a)
   (if (<= z 4.6e-13) (fma (fma z 1.0 (- y)) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -66.0) {
		tmp = x - a;
	} else if (z <= 4.6e-13) {
		tmp = fma(fma(z, 1.0, -y), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -66.0)
		tmp = Float64(x - a);
	elseif (z <= 4.6e-13)
		tmp = fma(fma(z, 1.0, Float64(-y)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -66.0], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.6e-13], N[(N[(z * 1.0 + (-y)), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -66 or 4.59999999999999958e-13 < z

   1. Initial program 92.1%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -66 < z < 4.59999999999999958e-13

   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

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 71.4

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

   10. Applied rewrites 71.4%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 71.0%

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

Alternative 11: 65.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-12}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(a, z, x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-12)
   (- x a)
   (if (<= z 4.8e-78) (fma a z x) (if (<= z 4.6e-16) (* y (- a)) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-12) {
		tmp = x - a;
	} else if (z <= 4.8e-78) {
		tmp = fma(a, z, x);
	} else if (z <= 4.6e-16) {
		tmp = y * -a;
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-12)
		tmp = Float64(x - a);
	elseif (z <= 4.8e-78)
		tmp = fma(a, z, x);
	elseif (z <= 4.6e-16)
		tmp = Float64(y * Float64(-a));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-12], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.8e-78], N[(a * z + x), $MachinePrecision], If[LessEqual[z, 4.6e-16], N[(y * (-a)), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.9999999999999998e-13 or 4.5999999999999998e-16 < z

   1. Initial program 92.2%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if -9.9999999999999998e-13 < z < 4.79999999999999999e-78

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 72.7

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

   10. Applied rewrites 72.7%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 64.2

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

   13. Applied rewrites 64.2%

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

   if 4.79999999999999999e-78 < z < 4.5999999999999998e-16

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 79.0

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

   7. Applied rewrites 79.0%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 60.4

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

   10. Applied rewrites 60.4%

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

   11. Taylor expanded in y around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

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

       4. lower-neg.f64 44.2

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

   13. Applied rewrites 44.2%

       \[\leadsto \color{blue}{\left(-a\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-12}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(a, z, x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-16}:\\ \;\;\;\;y \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -66:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(-y, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -66.0) (- x a) (if (<= z 4.6e-13) (fma (- y) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -66.0) {
		tmp = x - a;
	} else if (z <= 4.6e-13) {
		tmp = fma(-y, a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -66.0)
		tmp = Float64(x - a);
	elseif (z <= 4.6e-13)
		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, -66.0], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.6e-13], N[((-y) * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -66 or 4.59999999999999958e-13 < z

   1. Initial program 92.1%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -66 < z < 4.59999999999999958e-13

   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

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 92.4

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.9

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

   10. Applied rewrites 68.9%

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

Alternative 13: 66.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-12}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(a, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-12) (- x a) (if (<= z 1.3e-33) (fma a z x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-12) {
		tmp = x - a;
	} else if (z <= 1.3e-33) {
		tmp = fma(a, z, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-12)
		tmp = Float64(x - a);
	elseif (z <= 1.3e-33)
		tmp = fma(a, z, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-12], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.3e-33], N[(a * z + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.9999999999999998e-13 or 1.29999999999999997e-33 < z

   1. Initial program 92.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -9.9999999999999998e-13 < z < 1.29999999999999997e-33

   1. Initial program 97.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. frac-2neg N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

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

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

      14. distribute-frac-neg2 N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 71.2

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

   7. Applied rewrites 71.2%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 71.1

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

   10. Applied rewrites 71.1%

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

   11. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 59.5

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

   13. Applied rewrites 59.5%

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

Alternative 14: 60.1% 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 95.2%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 56.3

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

5. Applied rewrites 56.3%

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

Alternative 15: 17.1% 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 95.2%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 56.3

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

5. Applied rewrites 56.3%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 13.0

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

8. Applied rewrites 13.0%

   \[\leadsto \color{blue}{-a} \]
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 2024219 
(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))))