Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 93.0% → 97.7%

Time: 10.0s

Alternatives: 10

Speedup: 1.1×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 97.2

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

4. Applied rewrites 97.2%

   \[\leadsto x + \color{blue}{\frac{z}{t} \cdot \left(y - x\right)} \]
5. Add Preprocessing

Alternative 2: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z \cdot \left(y - x\right)}{t}\\ t_2 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* z (- y x)) t))) (t_2 (* z (/ (- y x) t))))
   (if (<= t_1 -5e+301) t_2 (if (<= t_1 5e+299) (fma (/ y t) z x) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((z * (y - x)) / t);
	double t_2 = z * ((y - x) / t);
	double tmp;
	if (t_1 <= -5e+301) {
		tmp = t_2;
	} else if (t_1 <= 5e+299) {
		tmp = fma((y / t), z, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(z * Float64(y - x)) / t))
	t_2 = Float64(z * Float64(Float64(y - x) / t))
	tmp = 0.0
	if (t_1 <= -5e+301)
		tmp = t_2;
	elseif (t_1 <= 5e+299)
		tmp = fma(Float64(y / t), z, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+301], t$95$2, If[LessEqual[t$95$1, 5e+299], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -5.0000000000000004e301 or 5.0000000000000003e299 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 75.2%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -5.0000000000000004e301 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 5.0000000000000003e299

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 80.3

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

   7. Applied rewrites 80.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{z \cdot \left(y - x\right)}{t} \leq -5 \cdot 10^{+301}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;x + \frac{z \cdot \left(y - x\right)}{t} \leq 5 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-22}:\\ \;\;\;\;-x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= t -8.2e-74)
     t_1
     (if (<= t 4.2e-173)
       (* (/ z t) y)
       (if (<= t 6.3e-22) (- (* x (/ z t))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (t <= -8.2e-74) {
		tmp = t_1;
	} else if (t <= 4.2e-173) {
		tmp = (z / t) * y;
	} else if (t <= 6.3e-22) {
		tmp = -(x * (z / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (t <= -8.2e-74)
		tmp = t_1;
	elseif (t <= 4.2e-173)
		tmp = Float64(Float64(z / t) * y);
	elseif (t <= 6.3e-22)
		tmp = Float64(-Float64(x * Float64(z / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t, -8.2e-74], t$95$1, If[LessEqual[t, 4.2e-173], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 6.3e-22], (-N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]), t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.20000000000000063e-74 or 6.3e-22 < t

   1. Initial program 86.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if -8.20000000000000063e-74 < t < 4.20000000000000003e-173

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 81.4

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

   4. Applied rewrites 81.4%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 74.1

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

   7. Applied rewrites 74.1%

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

   if 4.20000000000000003e-173 < t < 6.3e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.1%

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

   10. Applied rewrites 68.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-173}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{-22}:\\ \;\;\;\;-x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+46}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.1e-87)
   (fma (/ y t) z x)
   (if (<= y 4.1e+46) (- x (* x (/ z t))) (+ x (/ (* z y) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.1e-87) {
		tmp = fma((y / t), z, x);
	} else if (y <= 4.1e+46) {
		tmp = x - (x * (z / t));
	} else {
		tmp = x + ((z * y) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.1e-87)
		tmp = fma(Float64(y / t), z, x);
	elseif (y <= 4.1e+46)
		tmp = Float64(x - Float64(x * Float64(z / t)));
	else
		tmp = Float64(x + Float64(Float64(z * y) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.1e-87], N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[y, 4.1e+46], N[(x - N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.10000000000000007e-87

   1. Initial program 89.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 93.1

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 88.9

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

   7. Applied rewrites 88.9%

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

   if -2.10000000000000007e-87 < y < 4.1e46

   1. Initial program 92.5%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 91.5%

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

   if 4.1e46 < y

   1. Initial program 91.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 86.3

         \[\leadsto x + \frac{\color{blue}{y \cdot z}}{t} \]

   5. Applied rewrites 86.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-87}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+46}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+46}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= y -2.1e-87) t_1 (if (<= y 4.1e+46) (- x (* x (/ z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (y <= -2.1e-87) {
		tmp = t_1;
	} else if (y <= 4.1e+46) {
		tmp = x - (x * (z / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (y <= -2.1e-87)
		tmp = t_1;
	elseif (y <= 4.1e+46)
		tmp = Float64(x - Float64(x * Float64(z / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[y, -2.1e-87], t$95$1, If[LessEqual[y, 4.1e+46], N[(x - N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.10000000000000007e-87 or 4.1e46 < y

   1. Initial program 90.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if -2.10000000000000007e-87 < y < 4.1e46

   1. Initial program 92.5%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   7. Applied rewrites 91.5%

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

Alternative 6: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y t) z x)))
   (if (<= t -8.2e-74) t_1 (if (<= t 6.4e-106) (* (/ z t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / t), z, x);
	double tmp;
	if (t <= -8.2e-74) {
		tmp = t_1;
	} else if (t <= 6.4e-106) {
		tmp = (z / t) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / t), z, x)
	tmp = 0.0
	if (t <= -8.2e-74)
		tmp = t_1;
	elseif (t <= 6.4e-106)
		tmp = Float64(Float64(z / t) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[t, -8.2e-74], t$95$1, If[LessEqual[t, 6.4e-106], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.20000000000000063e-74 or 6.4e-106 < t

   1. Initial program 87.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if -8.20000000000000063e-74 < t < 6.4e-106

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 78.9

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 66.5

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

   7. Applied rewrites 66.5%

      \[\leadsto \color{blue}{y \cdot \frac{z}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-18}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3e-18) (* z (/ y t)) (/ (* z y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e-18) {
		tmp = z * (y / t);
	} else {
		tmp = (z * y) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3d-18)) then
        tmp = z * (y / t)
    else
        tmp = (z * y) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3e-18) {
		tmp = z * (y / t);
	} else {
		tmp = (z * y) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3e-18:
		tmp = z * (y / t)
	else:
		tmp = (z * y) / t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3e-18)
		tmp = Float64(z * Float64(y / t));
	else
		tmp = Float64(Float64(z * y) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3e-18)
		tmp = z * (y / t);
	else
		tmp = (z * y) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3e-18], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.99999999999999983e-18

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 28.8

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

   5. Applied rewrites 28.8%

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

   if -2.99999999999999983e-18 < t

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 39.0

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

   5. Applied rewrites 39.0%

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

   7. Applied rewrites 43.8%

      \[\leadsto \frac{z \cdot y}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 97.2

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

4. Applied rewrites 97.2%

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

Alternative 9: 40.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{z}{t} \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (/ z t) y))

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (z / t) * y

Julia

function code(x, y, z, t)
	return Float64(Float64(z / t) * y)
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 91.3%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 92.5

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

4. Applied rewrites 92.5%

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

5. Taylor expanded in y around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 40.4

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

7. Applied rewrites 40.4%

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

8. Final simplification 40.4%

   \[\leadsto \frac{z}{t} \cdot y \]
9. Add Preprocessing

Alternative 10: 37.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ z \cdot \frac{y}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* z (/ y t)))

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return z * (y / t)

Julia

function code(x, y, z, t)
	return Float64(z * Float64(y / t))
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.3%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 35.8

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

5. Applied rewrites 35.8%

   \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]
6. Add Preprocessing

Developer Target 1: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1805102239106601/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (* (/ z t) (- x y))) (if (< x 855006432740143/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z))))))

  (+ x (/ (* (- y x) z) t)))